Question

What is `int (sin x)/(cos^2x + 1) dx`?

Answer 1

`int\ (sin(x))/(cos^2(x)+1)\ dx=-arctan(cos(x))+C`

Explanation:

We will introduce a u-substitution with `u=cos(x)` . The derivative of `u` will then be `-sin(x)`, so we divide through by that to integrate with respect to `u`:

`int\ (sin(x))/(cos^2(x)+1)\ dx=int\ cancel(sin(x))/(1+u^2)*1/(-cancel(sin(x)))\ dx=-int\ 1/(1+u^2)\ du`

This is the familiar arctan integral, which means the result is:

`-int\ 1/(1+u^2)\ du=-arctan(u)+C`

We can resubstitute `u=cos(x)` to get the answer in terms of `x`:

`-arctan(cos(x))+C`