Integrating functions involving irreducible quadratics

Imagine we want to calculate the integral

$\int \frac1{x^2 + 4} dx$

We learned in the section on inverse trigonometric functions that the derivative of $\arctan{x}$ is given by

$\frac{d}{dx} \arctan{x} = \frac1{x^2 + 1}$

and so

$\int \frac1{x^2 + 1} dx = \arctan{x}$

There is a clear similarity between these two integrals, and we will see that we can use this result to solve the example integral, by using a small amount of algebraic manipulation and the Change of Variables Theorem.

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