Integral of square of Sine function

You can view as many worked out examples as you want. First you are shown the question. Click Show Answer to view the answer.
Click Show another Example to view another example. The best way to master mathematics is by practice. But practice requires time.
If you don't have the time to practice, you can always view many practice problems completely worked out and become good at it too.

$\int {\sin}^2\left(3x+1\right)\; dx$

**Answer**

$\int {\sin}^2\left(3x+1\right)\; dx$

Substitute $u=3x+1$. So, $du =3\; dx$

Or, $\frac{1}{3}\; du = dx$

And, $\int {\sin}^2\left(3x+1\right)\; dx = \frac{1}{3}\int {\sin}^2 u \;du$

$=\frac{1}{3}\left(\frac{1}{2}u-\frac{1}{4}\cos 2u \right) + C$

$=\frac{1}{6}u-\frac{1}{12}\cos 2u + C$

Substituting back $u=3x+1$

$=\frac{1}{6}\left(3x+1\right)-\frac{1}{12}\left( \cos 2\left(3x+1\right) \right) + C$

$=\frac{1}{2}x-\frac{1}{12}\cos\left(6x+2\right)+ C$