Problem 29 - Essential Calculus

Question

Find or evaluate the integral. (Complete the square, if necessary.) $$ \int \frac{x}{x^{4}+2 x^{2}+2} d x $$

Step-by-Step Solution

Answer

The solution to the integral is \(\frac{1}{2} \ln |x^{2}+1| - \frac{1}{2} \arctan (x^{2}+1) + C\).

Step 1: Complete the Square

Spotting that the denominator of the integrand is a perfect square, rewrite \(x^{4}+2x^{2}+2\) as \((x^2+1)^2+1\)

Step 2: Substitute the Completed Square

By substitution, Let \(u=x^{2}+1\). Then, \(du=2xdx\) or \(dx=du/2x\). Now the integral is \(\int \frac{xu}{u^2+1} \frac{du}{2x}\) which simplifies to \(\int \frac{u}{2(u^2+1)} du\)

Step 3: Solve the Integral

The integral is now in a form where you can use the arctangent integral. The result is \(\frac{1}{2} \int \frac{1}{u} du - \frac{1}{2} \int \frac{1}{u^{2}+1} du\). Solving further, the integral is \(\frac{1}{2} \ln |u| - \frac{1}{2} \arctan (u) + C\). Replace \(u=x^{2}+1\) to get the final answer.