Mathematics

The key is recognizing that the numerator is almost the derivative of the denominator. Notice that $\frac{d}{dx}(5x^2 + 1) = 10x$, so $2x$ is a scaled piece of that derivative. Split the numerator into a part proportional to the derivative of the denominator (which gives you a logarithm) and a constant part (which gives you an arctangent-like form). This transforms a tricky rational integral into two simpler pieces you already know how to handle.

The key is recognizing that the numerator is almost the derivative of the denominator. Notice that $\frac{d}{dx}(5x^2 + 1) = 10x$, so $2x$ is a scaled piece of that derivative. Split the numerator into a part proportional to the derivative of the denominator (which gives you a logarithm) and a constant part (which gives you an arctangent-like form). This transforms a tricky rational integral into two simpler pieces you already know how to handle.