Question

Trigonometric substitution: Some integrals that don’t start out involving trigonometric functions can be cleverly converted into ones that do, and then we can bring the techniques of this section to bear. In the next few problems this is how we will solve the integral

$\int \frac{dx}{x\sqrt{1-x^2}}$

Short answer

Answer is$dx=\cos udu$

Step-by-step solution

Step 1. Given information

An integral is$\int \frac{dx}{x\sqrt{1-x^2}}$

Step 2. Explanation

$\begin{array}{rcl}u& =& {\sin }^{-1}x\\ x& =& \sin u\\ dx& =& \cos udu\end{array}$