# Integral of $\frac{1}{x^{2} + 4}$

The calculator will find the integral/antiderivative of $\frac{1}{x^{2} + 4}$, with steps shown.

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Find $\int \frac{1}{x^{2} + 4}\, dx$.

### Solution

Let $u=\frac{x}{2}$.

Then $du=\left(\frac{x}{2}\right)^{\prime }dx = \frac{dx}{2}$ (steps can be seen here), and we have that $dx = 2 du$.

The integral can be rewritten as

$$\color{red}{\int{\frac{1}{x^{2} + 4} d x}} = \color{red}{\int{\frac{1}{2 \left(u^{2} + 1\right)} d u}}$$

Apply the constant multiple rule $\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$ with $c=\frac{1}{2}$ and $f{\left(u \right)} = \frac{1}{u^{2} + 1}$:

$$\color{red}{\int{\frac{1}{2 \left(u^{2} + 1\right)} d u}} = \color{red}{\left(\frac{\int{\frac{1}{u^{2} + 1} d u}}{2}\right)}$$

The integral of $\frac{1}{u^{2} + 1}$ is $\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$:

$$\frac{\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}{2} = \frac{\color{red}{\operatorname{atan}{\left(u \right)}}}{2}$$

Recall that $u=\frac{x}{2}$:

$$\frac{\operatorname{atan}{\left(\color{red}{u} \right)}}{2} = \frac{\operatorname{atan}{\left(\color{red}{\left(\frac{x}{2}\right)} \right)}}{2}$$

Therefore,

$$\int{\frac{1}{x^{2} + 4} d x} = \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}$$

$$\int{\frac{1}{x^{2} + 4} d x} = \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}+C$$
Answer: $\int{\frac{1}{x^{2} + 4} d x}=\frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}+C$