Question

Find the integral. \(\int \frac{x}{x^{4}+1} d x\)

Short answer

The integral \(\int \frac{x}{x^{4}+1}d x = \frac{1}{4}\ln |x^{4}+1|\)

Step-by-step solution

Identify the Substitution

Let \(u=x^{4}+1\). Then, the differential of \(u\) is \(du=4x^{3}dx\). Now, it's clear from the given integral that multiplying and dividing it by 4 doesn't change its value, so we transform the original integral \(\int \frac{x}{x^{4}+1}d x\) to \(\frac{1}{4} \int \frac{4x}{x^{4}+1}d x = \frac{1}{4} \int \frac{du}{u}\).

Perform Integration

The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln |u|\), therefore applying the integral gives us \(\frac{1}{4} \int \frac{du}{u} = \frac{1}{4}\ln |u|\) .

Substitute u back into the Integral

In step 1, we defined \(u=x^{4}+1\). We replace this back into our integral expression from step 2. Thus, our final answer will be \(\frac{1}{4}\ln |x^{4}+1|\).