Question

integration of sin^4(3t) dt

Answer 1

Even powers of sine and cosine are a little tricky.
You can either apply the `Half-Angle Identity` a few times,

Or you can apply the `Sine Reduction Formula`.

Half-Angle is probably the way to go on this one since it's only 4th power.

Answer 2

Half-Angle Identity for Sine,
\[\Large\bf\sf \sin^2x\quad=\quad \frac{1}{2}\left(1-\cos2x\right)\]

Answer 3

\[\Large\bf\sf \sin^43t\quad=\quad [\sin^23t]^2\quad=\quad \left[\frac{1}{2}\left(1-\cos6t\right)\right]^2\]

Answer 4

Understand how I applied it to this problem? :o