What is the integraton of Sin(2t)/1-cos(2t) dx ?

1 Answer
Feb 24, 2018

ln|sin(t)|+C

Explanation:

intsin(2t)/(1-cos(2t))dt

Recall the following identities:

sin(2t)=2sin(t)cos(t)

cos(2t)=1-2sin^2(t)

Rewrite the integral with the identities applied and simplify::

int(2sin(t)cos(t))/(1-(1-2sin^2(t))dt

int(cancel2sin(t)cos(t))/(cancel2sin^2(t))dt

int(cancelsin(t)cos(t))/(cancelsin(t)sin(t))dt

int(cos(t)/sin(t))dt

We can now use u-substitution:

u=sin(t)

du=cos(t)dt

cos(t)dt appears in the numerator of our integrand, so this substitution is valid:

int(du)/u

Integrate:

int(du)/u=ln|u|+C

Rewrite in terms of t:

ln|sin(t)|+C