Q. 67 - 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.)

$\int x^2{(x-1)}^{100}\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int x^2{(x-1)}^{100}\mathrm{d}x=\frac{1}{103}{(x-1)}^{103}+\frac{2}{102}{(x-1)}^{102}+\frac{1}{101}{(x-1)}^{101}+C$.

Step 1. Given Information

Solving the given integrals.

$\int x^2{(x-1)}^{100}\mathrm{d}x$

Step 2. Using the substitution method.

Let

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

In this question in which integration by substitution works even though the integrand is not at all in the form $f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)$. A clever change of variables will allow us to rewrite the integral so that it can be algebraically simplified.

$x=u+1$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int x^2{(x-1)}^{100}\mathrm{d}x=\int {(u+1)}^2{u}^{100}\mathrm{d}x \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\int {\{{\left(u\right)}^2+2\times u\times 1+(1)}^2\}{u}^{100}\mathrm{d}x \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\int (u^2+2u+1)u^{100}\mathrm{d}x \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\int (u^2·u^{100}+2u·u^{100}+1·u^{100})\mathrm{d}x \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\int (u^{102}+2u^{101}+u^{100})\mathrm{d}x \end{gathered}$$

Step 4. After simplifying.

$$\begin{gathered} \int x^2{(x-1)}^{100}\mathrm{d}x=\int u^{102}\mathrm{d}x+2\int u^{101}\mathrm{d}x+\int u^{100}\mathrm{d}x \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\left[\frac{u^{102+1}}{102+1}\right]+2\left[\frac{u^{101+1}}{101+1}\right]+\left[\frac{u^{100+1}}{100+1}\right]+C \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\left[\frac{u^{103}}{103}\right]+2\left[\frac{u^{102}}{102}\right]+\left[\frac{u^{101}}{101}\right]+C \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\frac{1}{103}{u}^{103}+\frac{2}{102}{u}^{102}+\frac{1}{101}{u}^{101}+C \\ \int x^2{(x-1)}^{100}\mathrm{d}x=\frac{1}{103}{(x-1)}^{103}+\frac{2}{102}{(x-1)}^{102}+\frac{1}{101}{(x-1)}^{101}+C \end{gathered}$$