Question

What is the integral of `int x/ cos^2(x)dx`?

Answer 1

`intx/(cos²(x))dx=xtan(x)+ln(|cos(x)|)+C`, `C in RR`

Explanation:

`intx/(cos²(x))dx=intx*1/(cos²(x))dx`
Using integration by parts :
`f(x)=x`
`f'(x)=1`
`g'(x)=1/(cos²x)`
`g(x)=tan(x)`
So: `intx/(cos²(x))dx=xtan(x)-inttan(x)dx`
`=xtan(x)-intsin(x)/cos(x)dx`
Let `X=cos(x)`
`dX=-sin(x)dx`
So:
`intx/(cos²(x))dx=xtan(x)+int1/XdX`
`intx/(cos²(x))dx=xtan(x)+ln(|X|)`
`intx/(cos²(x))dx=xtan(x)+ln(|cos(x)|)+C`, `C in RR`
\0/ here's our answer!