Q. 22 - 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.)

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

Step-by-Step Solution

Answer

The solution of the given integral is $\int \mathrm{x}{({\mathrm{x}}^2-1)}^2\mathrm{dx}=\frac{{({\mathrm{x}}^2-1)}^3}{6}+C$.

Step 1. Given Information

Solving the given integrals.
$\int x{(x^2-1)}^2\mathrm{d}x$

Step 2. Solving the given integral using substitution method.

Let

$$\begin{gathered} u=x^2-1 \\ \frac{du}{dx}=2x \\ du=2xdx \\ \frac{1}{2}du=xdx \end{gathered}$$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int x{(x^2-1)}^2\mathrm{d}x=\frac{1}{2}\int u^2\mathrm{du} \\ \int \mathrm{x}{({\mathrm{x}}^2-1)}^2\mathrm{dx}=\frac{1}{2}\left[\frac{{\mathrm{u}}^{2+1}}{2+1}\right]+C \\ \int \mathrm{x}{({\mathrm{x}}^2-1)}^2\mathrm{dx}=\frac{1}{2}\left[\frac{{\mathrm{u}}^3}{3}\right]+C \\ \int \mathrm{x}{({\mathrm{x}}^2-1)}^2\mathrm{dx}=\frac{{\mathrm{u}}^3}{6}+C \\ \int \mathrm{x}{({\mathrm{x}}^2-1)}^2\mathrm{dx}=\frac{{({\mathrm{x}}^2-1)}^3}{6}+C \end{gathered}$$