Question

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

$\int e^{u}du$

Short answer

The three integrals $\int (2x+1)e^{x^2+1}dx,\int \sin x{e}^{\cos x}dxand\int \frac{e^{\ln x}}{x}dx$will have form $\int e^{u}du$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.

$\int e^{u}du$

Step 2. 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

$\int (2x+1)e^{x^2+1}dx$

Step 3. 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

role="math" localid="1648747924788" $\int \sin x{e}^{\cos x}dx$

Step 4. 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

$\int \frac{e^{\ln x}}{x}dx$