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?)