Hello Blogmates.
This week in Calculus we learned trig sub and then we reviewed integration. Integration is something i really need help with becasue I can't seem to understand when to integrate regularly, bi part, substitute, or trig sub. Hints?
So, in order to help myself, i'll list a few formulas we should not forget.
S sinx = -cos x + c
S -sinx = cos x + c
S cosx = sin x + c
S tanx = ln /cosx/ + c
S secx = ln /secx + tanx/ + c
S cscx = - ln / secx + cotx/ + c
S cotx = ln/sinx/ + c
Next, it is very important that you learn these POWER REDUCTION FORMULAS:
cosx = 1/2 + 1/2cos2x
sinx = 1/2 - 1/2cox2x
And this PYTHAGOREAN IDENTITY:
cos^2x + sin^2x = 1
Now, i think the things i need help with the most is with choosing which integration method to do..hopefully that atleast comes with time?
The last thing i want to explain is a nice summary of trig sub.
So, first, you choose what box your problem is and find the information necessary; including: x, dx, sqrt, and whatever else is in the problem.
Second, you plug all of that back into the problem and hopefully cancel some things.
Third, integrate.
Fourth, form the triangle by the information given in the selected box.
Fifth, find the trig functions in the problem's answer by using SOHCAHTOA and the triangle.
Lastly, pray you got it right.
Now, a quote from my favorite youtube video, glozell,
PEACE AND BLESSINGS. PEACE AND BLESSINGS.
Sunday, September 12, 2010
Monday, September 6, 2010
Post #2
Kay. Since I've yet to touch on L'Hopital's Rule, I shall do so now.
The most important thing to learn about l'Hôpital's rule is when it should not be used:
Definitely do NOT use it when the limits of the two parts are not both 0, or both infinity. In this case the rule is likely to give a wrong answer!
Example:
limx->0+ (cos x)/x
is positive infinity, because the numerator approaches 1 while the denominator approaches 0. If we incorrectly apply l'Hôpital's rule, we get
limx->0+ (- sin x)/1 = 0.
So you DO use L'Hopital's Rule when you get an indeterminate in the first place...this is inf/inf, 0/0, etc.
Okay, for Trig SUB!!!!! I'm getting pretty good at this, so bear with me....
My trick is: Everytime I see a trig function to an odd power, I take out an even...After this I use an appropriate identity. It's really not all that hard...I have my notecards somewhere...just ask me for them..
OkAY!!!! for things you can comment on....
Does anyone know how to:
1. Divide stuff? like x^2 + x+ 7 all over x-8. the other day BRob tried to do a problem like that, and I failedddd miserably. Easier way??
2. Chasing the Rabbit. One time I ended up with chasing the rabbit, but the answer was something super easy. any hints as to when you should use by parts i.e. chasing the rabbit?
alright. night.
The most important thing to learn about l'Hôpital's rule is when it should not be used:
Definitely do NOT use it when the limits of the two parts are not both 0, or both infinity. In this case the rule is likely to give a wrong answer!
Example:
limx->0+ (cos x)/x
is positive infinity, because the numerator approaches 1 while the denominator approaches 0. If we incorrectly apply l'Hôpital's rule, we get
limx->0+ (- sin x)/1 = 0.
So you DO use L'Hopital's Rule when you get an indeterminate in the first place...this is inf/inf, 0/0, etc.
Okay, for Trig SUB!!!!! I'm getting pretty good at this, so bear with me....
My trick is: Everytime I see a trig function to an odd power, I take out an even...After this I use an appropriate identity. It's really not all that hard...I have my notecards somewhere...just ask me for them..
OkAY!!!! for things you can comment on....
Does anyone know how to:
1. Divide stuff? like x^2 + x+ 7 all over x-8. the other day BRob tried to do a problem like that, and I failedddd miserably. Easier way??
2. Chasing the Rabbit. One time I ended up with chasing the rabbit, but the answer was something super easy. any hints as to when you should use by parts i.e. chasing the rabbit?
alright. night.
blog 2
alright alright, so this week we had our first real test. it was kindaaaa hard, but i think i did well :) we are still working on integration .. by parts, trig sub, wallis formula, all that good stuff.
so, first i'll tell you what wallice's formula is (by the way idk how to spell it so ignore that)...
if you have an integration problem of sin or cos raised to a power.. this is when you use this
S cos^5(x)
alright, so if your degree is ODD, you do (2/3)(4/5)..(n-1/n) and simply multiply them together. so your answer would be 8/15.
S sin^8(x)
if your degree is EVEN, you do (1/2)(3/4)...(n-1/n) (pi/2). then multiply
so your answer would be (1/2)(3/4)(5/6)(7/8)(pi/2). i don't feel like multiplying it out haha.
HELP:
alright, trig sub. it's pretty much a bunch of formulas telling you what to substitute in and when to do it when you are integrating trig functions.
i know how to do these.. i just tend to mess up cuz i don't memorize when i have to do what. & i also didn't bring my book home to remember to put anything about it on here.. :x so.. could someone maybe go over a few formulas for me?
when to do synthetic division:
when your top function degree is larger than the top.
so say you had S (x^2 + 2x +5)/(x-6)
6 would go in your box, then 1, 2, 5...
i think. if i'm wrong someone please let me know! also, i get kinda lost after that.. i stop and don't remember what to do next, the only thing i remember is that i have to put my remainder over the bottom of the fraction at the end... help please
so, first i'll tell you what wallice's formula is (by the way idk how to spell it so ignore that)...
if you have an integration problem of sin or cos raised to a power.. this is when you use this
S cos^5(x)
alright, so if your degree is ODD, you do (2/3)(4/5)..(n-1/n) and simply multiply them together. so your answer would be 8/15.
S sin^8(x)
if your degree is EVEN, you do (1/2)(3/4)...(n-1/n) (pi/2). then multiply
so your answer would be (1/2)(3/4)(5/6)(7/8)(pi/2). i don't feel like multiplying it out haha.
HELP:
alright, trig sub. it's pretty much a bunch of formulas telling you what to substitute in and when to do it when you are integrating trig functions.
i know how to do these.. i just tend to mess up cuz i don't memorize when i have to do what. & i also didn't bring my book home to remember to put anything about it on here.. :x so.. could someone maybe go over a few formulas for me?
when to do synthetic division:
when your top function degree is larger than the top.
so say you had S (x^2 + 2x +5)/(x-6)
6 would go in your box, then 1, 2, 5...
i think. if i'm wrong someone please let me know! also, i get kinda lost after that.. i stop and don't remember what to do next, the only thing i remember is that i have to put my remainder over the bottom of the fraction at the end... help please
Sunday, September 5, 2010
Post #2
Hello my Calculus BC friends,
TRIG SUBSTITUTION!
Some basic integrals:
S sinu du = -cos u + C
S cosu du = sin u + C
S tan u du = -ln|cos u| + C
S cot u du = ln|sin u| + C
S secu du = ln|sec u + tan u| + C
S cscu du = -ln|csc u + cot u| + C
S sec^2 u du = tan u + C
S csc^2 u du = -cot u + C
Some identities:
sin^2x + cos^2x = 1 .
sin^2x = (1 - cos 2x)/2
cos^2x = (1 + cos 2x)/2
*What I try to do: usually try to take out some kind of squared, then change the to an identity, distribute in, and substitute.
*ALL the Rules:
SIN & COS guidelines:
1. If the power of the sine is odd and positive, save one sine
factor and convert the remaining factors to cosines. Then, expand
and integrate.
2. If the power of the cosine is odd and positive, save one cosine
factor and convert the remaining factors to sines. Then, expand
and integrate.
3. If the powers of both sine and cosine are even and
non negative, make repeated use of the half-angle identities for
sin^2x and cos^2x to convert the integrand to odd powers of the
cosine. Then proceed as in guideline 2.
SEC & TAN guidelines:
1. If the power of the secant is even and positive, save a secantsquared
factor and convert the remaining factors to tangents.
Then expand and integrate.
2. If the power of the tangent is odd and positive, save a secanttangent
factor and convert the remaining factors to secants. Then
expand and integrate.
3. If there are no secant factors and the power of the tangent is
even and positive, convert a tangent-squared factor to a secantsquared
factor, then expand and repeat if necessary.
4. If the integral is of the form S secmx dx, where m is odd and
positive, use integration by parts.
5. If none of the first four guidelines applies, try converting to
Wallis formula:
Only works with sin and cos when going from 0 to pi/2. n is the exponent
when n is ODD: (2/3)(4/5)(6/7)...(n -1)/n
EVEN: (1/2)(3/4)(5/6)...((n-1)/n)(pi/2)
HERE IS WHAT YOU CAN COMMENT ON:
Now I understand everything, but I somehow cannot always work the problems. Does anyone have some kind of trick on how to know when you look at a problem and know you have to either substitute, by part it, or trig sub? Also, do you know something that can help me remember how to do trig sub? (like the steps explained easier or a trick to remember or the steps you follow EVERY time?)
TRIG SUBSTITUTION!
Some basic integrals:
S sinu du = -cos u + C
S cosu du = sin u + C
S tan u du = -ln|cos u| + C
S cot u du = ln|sin u| + C
S secu du = ln|sec u + tan u| + C
S cscu du = -ln|csc u + cot u| + C
S sec^2 u du = tan u + C
S csc^2 u du = -cot u + C
Some identities:
sin^2x + cos^2x = 1 .
sin^2x = (1 - cos 2x)/2
cos^2x = (1 + cos 2x)/2
*What I try to do: usually try to take out some kind of squared, then change the to an identity, distribute in, and substitute.
*ALL the Rules:
SIN & COS guidelines:
1. If the power of the sine is odd and positive, save one sine
factor and convert the remaining factors to cosines. Then, expand
and integrate.
2. If the power of the cosine is odd and positive, save one cosine
factor and convert the remaining factors to sines. Then, expand
and integrate.
3. If the powers of both sine and cosine are even and
non negative, make repeated use of the half-angle identities for
sin^2x and cos^2x to convert the integrand to odd powers of the
cosine. Then proceed as in guideline 2.
SEC & TAN guidelines:
1. If the power of the secant is even and positive, save a secantsquared
factor and convert the remaining factors to tangents.
Then expand and integrate.
2. If the power of the tangent is odd and positive, save a secanttangent
factor and convert the remaining factors to secants. Then
expand and integrate.
3. If there are no secant factors and the power of the tangent is
even and positive, convert a tangent-squared factor to a secantsquared
factor, then expand and repeat if necessary.
4. If the integral is of the form S secmx dx, where m is odd and
positive, use integration by parts.
5. If none of the first four guidelines applies, try converting to
Wallis formula:
Only works with sin and cos when going from 0 to pi/2. n is the exponent
when n is ODD: (2/3)(4/5)(6/7)...(n -1)/n
EVEN: (1/2)(3/4)(5/6)...((n-1)/n)(pi/2)
HERE IS WHAT YOU CAN COMMENT ON:
Now I understand everything, but I somehow cannot always work the problems. Does anyone have some kind of trick on how to know when you look at a problem and know you have to either substitute, by part it, or trig sub? Also, do you know something that can help me remember how to do trig sub? (like the steps explained easier or a trick to remember or the steps you follow EVERY time?)
Post #2
Okay, so after what felt like the longest week ever, it's now time to do the blog. This week in Calculus BC I was kinda discouraged by trig sub because its something i really don't understand. I only do the problems right when i have the formulas in front of me...and after studying them for a week straight, and still getting mixed up on them, i'm finding it almost hopeless.
Hopefully someone can show me their study techniques?
But lets go over a few things...
For trig sub, something i always get wrong is WHEN to actually do the method..so, i believe it is when you can't basically bi-part something? correct?
Also, you should never bi-part or trig sub things when you only have the trig function and its derivative/ integral..just saying. I do it all the time and it is definately the hard way.
I really wish there is something i can explain that i know how to do, but there really isn't..
I guess i'll explain Wallis Formula.
So, you do this when you have sin or cos and the degree is EVEN:
1. Start with 1/2 and multiply the chronological numbers until you get to the exponent.
2. Then multiply by pi/two
3. Add +c
4. Box or circle your answer
When the degree is ODD:
1. Start with 1/2 and multiply the chronological numbers unitl you get to the exponent.
2. Put a + c
3. Box or circle your answer
So, i really feel like a baby and hopefully someone can help me..
i really just need all the helpful hints and basic problems explained to me.
i'm not quite sure why my brain hasn't kicked into school mode yet..
Hopefully someone can show me their study techniques?
But lets go over a few things...
For trig sub, something i always get wrong is WHEN to actually do the method..so, i believe it is when you can't basically bi-part something? correct?
Also, you should never bi-part or trig sub things when you only have the trig function and its derivative/ integral..just saying. I do it all the time and it is definately the hard way.
I really wish there is something i can explain that i know how to do, but there really isn't..
I guess i'll explain Wallis Formula.
So, you do this when you have sin or cos and the degree is EVEN:
1. Start with 1/2 and multiply the chronological numbers until you get to the exponent.
2. Then multiply by pi/two
3. Add +c
4. Box or circle your answer
When the degree is ODD:
1. Start with 1/2 and multiply the chronological numbers unitl you get to the exponent.
2. Put a + c
3. Box or circle your answer
So, i really feel like a baby and hopefully someone can help me..
i really just need all the helpful hints and basic problems explained to me.
i'm not quite sure why my brain hasn't kicked into school mode yet..
Sunday, August 29, 2010
Ryan's First Calculus BC Post!
This week in Calculus BC was pretty good.
Tuesday we re-learned Integration by-parts.
Integration by-parts:
Sudv = uv - Svdu
Example:
Sxe^(x)dx
Pick a u and dv and derive/integrate:
u = x du = 1 dx
dv = e^x(dx) v = e^x
You then plug these into your equation:
xe^(x) - Se^(x)dx
And solve:
xe^(x) - e^(x) + C
We also reviewed "Chasing the Rabbit"
This method of integration happens when you have an e term and a trig term in an integration problem and you basically keep integrating by-parts until you arrive at an integration term the same as you started off with, you them set everything you have equal to the original problem.
We learned something brand new this week that I'm having a little trouble with: Trig Integration.
I get the basic problems, but I was having issues with the homework over the weekend.
Two examples would be:
Ssec^3(pix)dx
and
Stan^5(x/2)dx
Tuesday we re-learned Integration by-parts.
Integration by-parts:
Sudv = uv - Svdu
Example:
Sxe^(x)dx
Pick a u and dv and derive/integrate:
u = x du = 1 dx
dv = e^x(dx) v = e^x
You then plug these into your equation:
xe^(x) - Se^(x)dx
And solve:
xe^(x) - e^(x) + C
We also reviewed "Chasing the Rabbit"
This method of integration happens when you have an e term and a trig term in an integration problem and you basically keep integrating by-parts until you arrive at an integration term the same as you started off with, you them set everything you have equal to the original problem.
We learned something brand new this week that I'm having a little trouble with: Trig Integration.
I get the basic problems, but I was having issues with the homework over the weekend.
Two examples would be:
Ssec^3(pix)dx
and
Stan^5(x/2)dx
Steph's First Calc BC Post
So, lets get this thing going again. First i'll give a little summary of things I did this past week. We went over lo'hospital's rule, integration, substitution, bi-parts, trig integration, and i think there's one more thing that just isn't clicking this moment.
So since this week was basically a review, let me touch on everything we did. Kinda like a re-review.
So, lets do some reviewing...
LO'HOSPITAL'S RULE:
1. This deals only with limits..so if you don't have a limit, dont do this!
2. Plug in the number the limit is approaching into the equation and make sure you get an infinitave..such as infinity/infinity, zero/zero, infinity/zero, zero/infinity.
3. Next, you need to take the derivative of the top, and the derivitave of the bottom.
******NOT QUOTIENT RULE, JUST TWO DERIVATIVES.
4. Plug in the number the limit is approaching
5. If you get another infinitive, repeat.
INTEGRATION:
*So, i know this is a little review, but don't laugh...you mess up on these sometimes too!
1. You know it's an integral when you see the "s" looking thinggyy.
2. Add one to the exponent
3. Multiply the coefficient by the recriprocal of the new exponent.
If indefinite, be sure to include +c, if not, solve.
So, these are the two topics that i was most confortable with my quizzes.
Now, the things i'm pretty sure i didn't do good on, bi-parts and trig substitution..
These two topics just don't click in my head. Any suggestions on how i should study thesee things?
Also, how do you tell if its biparts or trig sub..?
What would you do for S xarctanx?
So since this week was basically a review, let me touch on everything we did. Kinda like a re-review.
So, lets do some reviewing...
LO'HOSPITAL'S RULE:
1. This deals only with limits..so if you don't have a limit, dont do this!
2. Plug in the number the limit is approaching into the equation and make sure you get an infinitave..such as infinity/infinity, zero/zero, infinity/zero, zero/infinity.
3. Next, you need to take the derivative of the top, and the derivitave of the bottom.
******NOT QUOTIENT RULE, JUST TWO DERIVATIVES.
4. Plug in the number the limit is approaching
5. If you get another infinitive, repeat.
INTEGRATION:
*So, i know this is a little review, but don't laugh...you mess up on these sometimes too!
1. You know it's an integral when you see the "s" looking thinggyy.
2. Add one to the exponent
3. Multiply the coefficient by the recriprocal of the new exponent.
If indefinite, be sure to include +c, if not, solve.
So, these are the two topics that i was most confortable with my quizzes.
Now, the things i'm pretty sure i didn't do good on, bi-parts and trig substitution..
These two topics just don't click in my head. Any suggestions on how i should study thesee things?
Also, how do you tell if its biparts or trig sub..?
What would you do for S xarctanx?
first calc BC post..
alright, so back to another year of blogs... YAY :D
haha, anyways. this week we took 3 tests. oh my lord, yes i know.
one on l'hopital's rule, one on basic integration, and one on integration by parts. i did great on the first one, hopefully i did good on the other two, too. :)
ok, so let's go over some integration by parts.
first you need to find your u and your dv.
usually your u is whatever can be reduced.
then after that, you find your du and your v.
*remember sometimes your dv can be your dx and your v would then become x.
then you simply plug into the formula and integrate.
oh .. & the formula is
uv - S vdu
*p.s. - S is the integration symbol
EXAMPLE:
S xsin2x dx
u = x v = -1/2cos2x
du = 1 dx dv=sin2x
x-1/2cos2x - S -1/2cos2x
=> your answer would be...
-(x)1/2cos(2x) + 1/4sin(2x) + c
easy right? yeah. just wait til you get to the hard ones. lol OH & there is chasing the rabbit. this happens whenever you integrate something using by parts twice, and you end up with the same thing you started with in the original problem. then you simply set it equal to the original and solve it like that.
Another aspect of calculus bc that we reviewed this week, which i'll review briefly, is l'hopitals rule.
this is whenever you try to find a limit of something, but it's in indeterminate form... then you have to use l'hopitals rule.
which means you take the derivative of the top of the fraction and the derivative of the bottom of the fraction... SEPARATELY! no quotient rule. then plug in & solve.
but, you must make sure that it is in fraction form.. because you cannot use l'hopitals rule if it is not.
alright, well that's all for now.
haha, anyways. this week we took 3 tests. oh my lord, yes i know.
one on l'hopital's rule, one on basic integration, and one on integration by parts. i did great on the first one, hopefully i did good on the other two, too. :)
ok, so let's go over some integration by parts.
first you need to find your u and your dv.
usually your u is whatever can be reduced.
then after that, you find your du and your v.
*remember sometimes your dv can be your dx and your v would then become x.
then you simply plug into the formula and integrate.
oh .. & the formula is
uv - S vdu
*p.s. - S is the integration symbol
EXAMPLE:
S xsin2x dx
u = x v = -1/2cos2x
du = 1 dx dv=sin2x
x-1/2cos2x - S -1/2cos2x
=> your answer would be...
-(x)1/2cos(2x) + 1/4sin(2x) + c
easy right? yeah. just wait til you get to the hard ones. lol OH & there is chasing the rabbit. this happens whenever you integrate something using by parts twice, and you end up with the same thing you started with in the original problem. then you simply set it equal to the original and solve it like that.
Another aspect of calculus bc that we reviewed this week, which i'll review briefly, is l'hopitals rule.
this is whenever you try to find a limit of something, but it's in indeterminate form... then you have to use l'hopitals rule.
which means you take the derivative of the top of the fraction and the derivative of the bottom of the fraction... SEPARATELY! no quotient rule. then plug in & solve.
but, you must make sure that it is in fraction form.. because you cannot use l'hopitals rule if it is not.
alright, well that's all for now.
Abbey's first BC blog!
Well first 2 week of school done!
One thing I'm comfortable with is L'Hopital's Rule. It is used when an indeterminate form occurs with a limit. It could be in any form of: infinity-infinity, 0/0, infinity/infinity, 0(infinity), 0^0, 1^infinity, and infinity^0.
When you get an indeterminate form, you take the derivative top then derivative of bottom. If you still get an indeterminate form just repeat the second step.
EXAMPLES:
lim e^2x - 1/x = e^2(0) - 1/1 = 1-1/0 = o/o <---indeterminate form
x-->0
So, take derivative of top then bottom.
e^2x - 1/x = 2e^2x/1 = 2(1)/1 = 2
lim lnx/x = infinity/infinity <---indeterminate form
x-->infinity
1/x/1 = 1/x = 0
With this example you would use your limit rules because it is as x approaches infinity. Since the degree of bottom is larger than the degree of top it equal zero.
Another thing we have covered is substitution. I can usually pick out my u and du, but sometimes I have trouble completing the problem. I guess I just need more practice.
EXAMPLE:
S tsint^2dt
u=t^2 du=2t
1/2 S sinudu
-1/2cost^2 +C
By parts is also something I can usually get my u, du, dv, and v. I just sometimes have a hard time finishing the problem. The formula is: uv - Svdu. Remember that the u is something that you want to reduce and dv can be the dx.
EXAMPLE:
S xe^x
u=x v=e^x
du=1 dv=e^xdx
xe^x- S e^xdx
xe^x-e^x +C
What I need help on! The whole sin, cos, secant, tan stuff with the formulas. I really didn't get the homework...
How would you work this?
S sec^4 5xdx
One thing I'm comfortable with is L'Hopital's Rule. It is used when an indeterminate form occurs with a limit. It could be in any form of: infinity-infinity, 0/0, infinity/infinity, 0(infinity), 0^0, 1^infinity, and infinity^0.
When you get an indeterminate form, you take the derivative top then derivative of bottom. If you still get an indeterminate form just repeat the second step.
EXAMPLES:
lim e^2x - 1/x = e^2(0) - 1/1 = 1-1/0 = o/o <---indeterminate form
x-->0
So, take derivative of top then bottom.
e^2x - 1/x = 2e^2x/1 = 2(1)/1 = 2
lim lnx/x = infinity/infinity <---indeterminate form
x-->infinity
1/x/1 = 1/x = 0
With this example you would use your limit rules because it is as x approaches infinity. Since the degree of bottom is larger than the degree of top it equal zero.
Another thing we have covered is substitution. I can usually pick out my u and du, but sometimes I have trouble completing the problem. I guess I just need more practice.
EXAMPLE:
S tsint^2dt
u=t^2 du=2t
1/2 S sinudu
-1/2cost^2 +C
By parts is also something I can usually get my u, du, dv, and v. I just sometimes have a hard time finishing the problem. The formula is: uv - Svdu. Remember that the u is something that you want to reduce and dv can be the dx.
EXAMPLE:
S xe^x
u=x v=e^x
du=1 dv=e^xdx
xe^x- S e^xdx
xe^x-e^x +C
What I need help on! The whole sin, cos, secant, tan stuff with the formulas. I really didn't get the homework...
How would you work this?
S sec^4 5xdx
Mal's First BC Post...
So, I've got to get back into the routine of doing these things. Luckily my dad's got my back and remembered for me.
Calculus. Right. Here I go...
A brief overview of basic integration (a few tips):
1. Remember that normally your u is whatever is being raised to a power or under the square root or the bottom of a fraction, etc. For example:
∫x/√(x^2+1)
u=x^2+1
du=2x
Now balance the 2 by putting a ½ in front and you have:
1/2∫du/(√u)
=√(x+1) + C
2. You have to be able to recognize basic derivatives (i.e. trig ones) So:
∫secx tanx
We know that the derivative of sec x is sec tan, so obviously the integral equals:
secx+C
3. Also remember that if you see 1/x and your integrating..that’s natural log integration.
4. Okay. By parts. You must remember that the formula is as follows:
uv- ∫(v)du
That, my friends is a crucial part. Also remember that your u is usually that which will decrease more. So if you have x and x^3, your u would be x and dv x^3. Got it? Good. HOWEVER, if you have a ln in the equation, that needs to be your u because you cannot integrate a ln…It’s just not conducive.
5. Next subject: Trig Substitution. I know we had some homework over the weekend, and for a lot of it, I was unsure. I did find out, I believe, that the integral of sec x is always going to be:
∫secx = ln(secx + tanx) + C
Don’t ask me why, that’s just what I found. You have to memorize it I believe? Now my main problem is trying to figure out when to substitute in certain trig identities. Does anyone know a way to remember it ? In desperate need of help if I’m going to pass that quiz on Wednesday. Muchas Gracias!
Calculus. Right. Here I go...
A brief overview of basic integration (a few tips):
1. Remember that normally your u is whatever is being raised to a power or under the square root or the bottom of a fraction, etc. For example:
∫x/√(x^2+1)
u=x^2+1
du=2x
Now balance the 2 by putting a ½ in front and you have:
1/2∫du/(√u)
=√(x+1) + C
2. You have to be able to recognize basic derivatives (i.e. trig ones) So:
∫secx tanx
We know that the derivative of sec x is sec tan, so obviously the integral equals:
secx+C
3. Also remember that if you see 1/x and your integrating..that’s natural log integration.
4. Okay. By parts. You must remember that the formula is as follows:
uv- ∫(v)du
That, my friends is a crucial part. Also remember that your u is usually that which will decrease more. So if you have x and x^3, your u would be x and dv x^3. Got it? Good. HOWEVER, if you have a ln in the equation, that needs to be your u because you cannot integrate a ln…It’s just not conducive.
5. Next subject: Trig Substitution. I know we had some homework over the weekend, and for a lot of it, I was unsure. I did find out, I believe, that the integral of sec x is always going to be:
∫secx = ln(secx + tanx) + C
Don’t ask me why, that’s just what I found. You have to memorize it I believe? Now my main problem is trying to figure out when to substitute in certain trig identities. Does anyone know a way to remember it ? In desperate need of help if I’m going to pass that quiz on Wednesday. Muchas Gracias!
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