Question

What is `int sin^3(12x)cos^2(12x) dx`?

Answer 1

`(3cos^5(12x)-5cos^3(12x))/180+ C`

Explanation:

Whenever you have sine and cosine functions in the integral, and the term inside terms are the same, its a good idea to check if you can use the identity;

`sin^2x + cos^2x=1`

In this case, we can rearrange so that the expression reads;

`sin^2(12x)=1-cos^2(12x)`

Now we can rewrite the integral a little bit.

`int sin^3(12x)cos^2(12x)dx`

`=int sin(12x)sin^2(12x)cos^2(12x)dx`

`=int sin(12x)(1-cos^2(12x))cos^2(12x)dx`

At first glance, this looks messier, but now we can use substitution to make the integration easier.

Let `u=cos(12x)`

Using the chain rule gives us;

`(du)/(dx) = -12sin(12x)`

`-1/12 du = sin(12x)dx`

Now that we have expressions for `u` and `du`, lets take another look at the intgral.

`int color(green)sin(12x)color(red)((1-cos^2(12x))cos^2(12x))color(green)dx`

We can make the `du` substitution for the green part, and the `u` substitutions in the red part.

`int -1/12(1-u^2)u^2du`

We can move the constant out of the integral, and multiply `u^2` through the parenthesis, to get the integral;

`-1/12 int u^2-u^4 du`

Using the power rule, this integral is not so bad.

`-1/12 (u^3/3-u^5/5) + C`

Multiply the `-1/12` through.

`u^5/60-u^3/36 +C`

Now we can find a common denominator.

`(3u^5-5u^3)/180 +C`

Re-substituting for `u` we get;

`(3cos^5(12x)-5cos^3(12x))/180+ C`