Question

Solve integral `\intx^p/(x\sqrt(1-x^2))dx`?

I recognize that `\int1/\sqrt(1-x^2)` involves trigonometric substitution, but I don't know how to take the other `x`'s out

Answer 1

`int \ (x^p)/(xsqrt(1-x^2)) \ dx = int \ sin^(p-1) theta \ d theta`

And:

`int \ sin^nx \ dx = -1/nsin^(n-1)x \ cosx + (n-1)/n \ int \ sin^(n-2)x \ dx`

Explanation:

We seek:

`I = int \ (x^p)/(xsqrt(1-x^2)) \ dx`

Consider:

`I_n =int \ x^n/(sqrt(1-x^2)) \ dx`

We can perform a substitution:

`x = sin theta => dx/(d theta) = cos theta`

And if we substitute into the integral, we get:

`I_n = int \ (sin theta)^n/(sqrt(1-(sin theta)^2)) \ cos theta \ d theta`

`\ \ \ = int \ (sin^n theta)/(sqrt(1 - sin^2 theta)) \ cos theta \ d theta`

`\ \ \ = int \ (sin^n theta)/(sqrt(cos^2 theta)) \ cos theta \ d theta`

`\ \ \ = int \ sin^n theta \ d theta`

Then we can write:

`I = int \ (x^(p-1))/(sqrt(1-x^2)) \ dx`

`\ \ = I_(p-1)`

`\ \ = int \ sin^(p-1) theta \ d theta`

And, a reduction formula can be used for the resultant integral.

`int \ sin^nx \ dx = -1/nsin^(n-1)x \ cosx + (n-1)/n \ int \ sin^(n-2)x \ dx`