Q. 21 - Calculus

Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int {(3x+1)}^2\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int {(3x+1)}^2\mathrm{d}x=\frac{{(3x+1)}^3}{9}+C$.

Step 1. Given Information

Solving the given integrals.

$\int {(3x+1)}^2\mathrm{d}x$

Step 2. Solving the given integral using substitution method.

Let

$$\begin{gathered} u=3x+1 \\ \frac{du}{dx}=3 \\ du=3dx \\ \frac{1}{3}du=dx \end{gathered}$$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int {(3x+1)}^2\mathrm{d}x=\frac{1}{3}\int u^2\mathrm{du} \\ \int {(3x+1)}^2\mathrm{d}x=\frac{1}{3}\left[\frac{u^{2+1}}{2+1}\right]+C \\ \int {(3x+1)}^2\mathrm{d}x=\frac{1}{3}·\frac{u^3}{3}+C \\ \int {(3x+1)}^2\mathrm{d}x=\frac{u^3}{9}+C \\ \int {(3x+1)}^2\mathrm{d}x=\frac{{(3x+1)}^3}{9}+C \end{gathered}$$