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

1 Answer
Steve M
Jun 2, 2018

# 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 #

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