Question

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

$\int \sin udu$

Short answer

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

Step-by-step solution

Step 1. Given Information 

For given integral we have to write down three integrals that will have that form after a substitution of variables.
$\int \sin udu$

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

Let $u=x^2$

$$\begin{gathered} \frac{du}{dx}=2x \\ du=2xdx \end{gathered}$$

This substitution changes the integral into

localid="1648748046608" $\int 2x\sin x^{2}dx$

Step 3. The second 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

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

Step 4. The third 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

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