Trig Integrals

sinx to odd power / cosx to even power

u = cosx

du = -sinxdx

dx = du/-sinx

Plugging the dx in will subtract one power from the odd sinx to make it even, allowing the use of sin²x = 1 - cos²x so everything will be in terms of u

cosx to odd power / sinx to even power

u = sinx

du = cosdx

dx = du/cos

Plugging the dx in will subtract one power from the odd cosx to make it even, allowing the use of cos²x = 1 - sin²x so everything will be in terms of u

secx to even power

u = tanx

du = sec²xdx

dx = du/sec²x

Plugging the dx in will subtract two powers from the even secx to make it even, allowing the use of sec²x = tan²x + 1 so everything will be in terms of u

cscx to even power

u = cotx

du = -csc²xdx

dx = du/-csc²x

Plugging the dx in will subtract two powers from the even cscx to make it even, allowing the use of csc²x = cot²x + 1 so everything will be in terms of u

tanx to odd power

u = secx

du = secxtanxdx

dx = du/secxtanx

Plugging the dx in will subtract one power from the odd tanx to make it even, allowing the use of tan²x = sec²x - 1 so everything will be in terms of u

cotx to odd power

u = cscx

du = -cscxcotxdx

dx = du/-cscxcotx

Plugging the dx in will subtract one power from the odd cotx to make it even, allowing the use of cot²x = csc²x - 1 so everything will be in terms of u

cos to even power / sin to even power

use cos²x = (1 + cos2x) / 2 and sin²x = (1 - cos2x) / 2

tanx to even power / secx to odd power

pray and use trig identities

cotx to even power / cscx to odd power

pray and use trig identities

sin²x + cos²x = 1

yep

sin²x =

(1 - cos²x) / 2

cos²x =

(1 + cos²x) / 2