Q. 21 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The solution is: $\frac{3}{4}\ln \left|1+4x\right|+C$

Step 1: Given

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

Step 2: To Find

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

Step 3: Calculation

Let, 

$$\begin{gathered} u=1+4x \\ du=4dx \\ \frac{du}{4}=dx \end{gathered}$$

Using substitution we get:

$$\begin{gathered} \int \frac{3}{1+4x}dx \\ =\int \frac{3}{u}\frac{du}{4} \\ =\frac{3}{4}\int \frac{du}{u} \\ =\frac{3}{4}\ln \left|u\right|+C \\ =\frac{3}{4}\ln \left|1+4x\right|+C \end{gathered}$$

Where, $C$= integrating constant.