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.
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I'm still iffy on when to trig sub it, by part it, substitute it, or integrate it. But with all this practice I'm learning a few things.
ReplyDeleteNow remember from your formula sheet that if you just have sinx, cosx, tanx, cotx, secx, cscx, csc^2x, sec^2x, cscxcotx, secxtanx these integrate regularly.
Now for substitution that is tricky because my brain really doesn't think about well the derivative is this so..blah blah. But some are really easy to see so just start to think of that first. Also, remember sec and tan, cos and sin are opposites so that works in substitution.
By parts sometimes has an e or like x^2 with a trig function. If substitution fails, just try by parts.
If you have like sin or cos or tan and are have odd powers or sec with even power think take out something^2 and use identities.
If you got a square root trig sub with that triangle.
And don't forget sec^3x is a formula! NOT BY PARTS
What I do when I see a problem that involves trigonometry functions is try to get it in simplest forms, changing things like cot to cos/sin and sin2x to 2sinxcosx. I usually do that and it helps point things out. Even if you can't integrate once you change things, at least you know that you ruled one thing out.
ReplyDeleteIf you see say tan...I always break that up into sine and cosine. Also. most of the time, you can get something down to where you can just use the power reduction formula and then the double angle formula.
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