Question

Integrate 1/[(x^2) + 1] dx

Answer 1

Let \(x=\tan u\), then \(dx=\sec^2u~du\):
\[\int\frac{\sec^2u}{\tan^2u+1}~du\]
Recall the identity \(\tan^2+1=\sec^2u\):
\[\int\frac{\sec^2u}{\sec^2u}~du=\int du\]

Answer 2

\[\int\limits \frac{1}{x^2+1} \, dx=\tan ^{-1}(x) \]