Q. 6 - Calculus

Question

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

$\int u^{2}du$

Step-by-Step Solution

Answer

The three integrals $\int \sin x{\left(\cos x\right)}^{2}dx,\int \frac{{\left(\ln x\right)}^2}{x}dx \mathrm{and} \int (2x+1)(x^2+x)dx$ will have form $\int u^{2}du$ after a substitution of variables.

Step 1. Given Information

For given integral we have to write down three integrals that will have that form after a substitution of variables.
$\int u^{2}du$

Step 2. The first integrals that will have that form after a substitution of variables.

Let 

$$\begin{gathered} u=\cos x \\ \frac{du}{dx}=\sin x \\ du=\sin xdx \end{gathered}$$

This substitution changes the integral into 

$\int \sin x{\left(\cos x\right)}^{2}dx$

Step 3. The first integrals that will have that form after a substitution of variables.

Let 

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

This substitution changes the integral into 

$\int \frac{{\left(\ln x\right)}^2}{x}dx$

Step 4. The first integrals that will have that form after a substitution of variables.

Let

$$\begin{gathered} u=x^2+x \\ \frac{du}{dx}=2x+1 \\ du=(2x+1)dx \end{gathered}$$

This substitution changes the integral into 

$\int (2x+1)(x^2+x)dx$