Question

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.

$\int \frac{3}{1+4x^2}dx$

Short answer

The solution is: $\frac{3}{2}{\tan }^{-1}\left(2x\right)+C$

Step-by-step solution

Given

The given integral is: $\int \frac{3}{1+4x^2}dx$

To Find 

Use integration formulas, algebra, and educated guess-and-check strategies to find the given integral.

Calculation

Let,

$$\begin{gathered} u=2x \\ du=2dx \\ \frac{du}{2}=dx \end{gathered}$$

Using substitution we get:

role="math" localid="1648934584019" $$\begin{gathered} \int \frac{3}{1+4x^2}dx \\ =\int \frac{3}{1+{\left(2x\right)}^2}dx \\ =\int \frac{3}{1+u^2}\frac{du}{2} \\ =\frac{3}{2}\int \frac{1}{1+u^2}du \\ =\frac{3}{2}{\tan }^{-1}\left(u\right)+C \\ =\frac{3}{2}{\tan }^{-1}\left(2x\right)+C \end{gathered}$$

Where, C = integrating constant.