Question

How do you integrate `int x^3 t an x dx` using integration by parts?

Answer 1

`intx^3tan(x)dx = ln|cos(x)|(x^3 - 3x^2 + 6x - 6 + c)`

Explanation:

Say `dv = tan(x)` so `v = ln|cos(x)|`
And `u = x^3` so `du = 3x^2`

`intx^3tan(x)dx = x^3*ln|cos(x)| - 3inttan(x)*x^2dx`

For the latter integral, repeat

`dv = tan(x)` so `v = ln|cos(x)|`, `u = x^2` so `du = 2x`

`intx^3tan(x)dx = x^3*ln|cos(x)| - 3(x^2ln|cos(x)| - 2inttan(x)xdx)`
`intx^3tan(x)dx = x^3*ln|cos(x)| - 3x^2ln|cos(x)| + 6inttan(x)xdx`

And once more to finish it

`dv = tan(x)` so `v = ln|cos(x)|`, `u = x` so `du = 1`

`intx^3tan(x)dx = x^3ln|cos(x)| - 3x^2ln|cos(x)| + 6(xln|cos(x)| - ln|cos(x)| + c)`

`intx^3tan(x)dx = x^3ln|cos(x)| - 3x^2ln|cos(x)| + 6xln|cos(x)| - 6ln|cos(x)| + c`

Or

`intx^3tan(x)dx = ln|cos(x)|(x^3 - 3x^2 + 6x - 6 + c)`