Question

What is the integral of `int tan^5(x)*sec^4(x)dx`?

Answer 1

`= 1/8tan^8(x) + 1/6tan^6(x) + C`

Explanation:

`int dx qquad tan^5(x)*sec^4(x)`

`= int dx qquad tan^5(x)*color{red}{sec^2(x)}*sec^2(x)`

using well known identity....
`= int dx qquad tan^5(x)*color{red}{(tan^2(x) + 1)}*sec^2(x)`

`= int dx qquad (tan^7(x) + tan^5(x))*sec^2(x)`

`= int dx qquad tan^7(x)*sec^2(x) + tan^5(x)*sec^2(x)`

`= 1/8tan^8(x) + 1/6tan^6(x) + C`