Problem 21 - Essential Calculus

Question

Find the integral. $$ \int \frac{\sqrt{1-x^{2}}}{x^{4}} d x $$

Step-by-Step Solution

Answer

The integral of the given function is \( -|x| - 1 \) .

Step 1: Apply Substitution

A good choice for a substitution is \(u = x^{2}\), so \( du = 2xdx \). Also the integral changes to \( \frac{1}{2} \int \frac{\sqrt {1 - u}}{u^{2}} du \).

Step 2: Simplify the integral

Reorganize the integral function into a form comfortable for resolving. The integral now becomes \( -\frac{1}{2} \int \frac{u'^{1/2}}{u} du \).

Step 3: Apply Power rule for integration

Now we can easily find the integral using a standard rule. This requires recognizing one part of the function as the derivative of another part. The integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). The integral thus becomes: \( -\frac{1}{2}(2\sqrt{u} + 2) = -\sqrt{u} - 1 \).

Step 4: Substitute back u

Now we must revert to the original variable, x. Replacing u by \(x^{2}\), we find: The integral equals \( -\sqrt{x^{2}} - 1 = -|x| - 1\).