Question

What is the the value of `int sqrt(tanx/(sinxcosx)) dx`?

Answer 1

`= ln|secx + tanx| + C`

Explanation:

Start by simplifying the expression within the integral.

`=intsqrt((sinx/cosx)/(sinxcosx))dx`

`=intsqrt((sinx)/(cosxsinxcosx))dx`

`=intsqrt(1/cos^2x)dx`

`=int(1/cosx)dx`

`=int(secx)dx`

This is a tricky integral to do. Multiply everything by `secx+ tanx`.

`=int(secx xx (secx + tanx)/(secx + tanx))dx`

`=int(sec^2x + secxtanx)/(secx + tanx)dx`

Let `u = secx + tanx`. Then `(du)/dx = sec^2x + secxtanx` and `du = sec^2x + secxtanx dx`. Substitute:

`=int(du)/u`

`=ln|u| + C`

`= ln|secx + tanx| + C`

Hopefully this helps!