Question

How do you integrate `int xtan(x^2)sec(x^2)` using substitution?

Answer 1

`1/2sec(x^2)+C`

Explanation:

`I=intxtan(x^2)sec(x^2)dx`

The first substitution we will make is `u=x^2`. Differentiating both sides gives us `(du)/dx=2x`, so `du=2xdx`. Notice that in our integrand we currently have `xdx`, so multiply the interior by `2` and the exterior by `1/2` to balance ourselves out.

`I=1/2int2xtan(x^2)sec(x^2)dx`

`I=1/2inttan(x^2)sec(x^2)(2xdx)`

Substituting in our values for `u` and `du`:

`I=1/2inttan(u)sec(u)du`

This is the integral for `sec(u)`, since `d/(du)sec(u)=sec(u)tan(u)`.

`I=1/2sec(u)+C`

Since `u=x^2`:

`I=1/2sec(x^2)+C`