Question

How do you find the integral of `tan (x) sec^3(x) dx`?

Answer 1

The required integral is

`I = int tanxsec^3xdx`

which can be written as

`I = intsec^2x(secxtanx)dx`

Let `secx = t`

Differentiating with respect to t,

`secxtanxdx/dt = 1`
`=> secxtanxdx = dt`

This gives us the integral

`I = intt^2dt`

which has a trivial solution of

`I = t^3/3 + C`

Replacing the value of `t = secx`, the final answer is

`I = int tanxsec^3xdx = sec^3x/3 + C`