Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int x\sinh x^2\mathrm{d}x$

Short answer

The solution of the given integral is $\int x\sinh x^2\mathrm{d}x=\frac{1}{2}\cosh x^2+C$.

Step-by-step solution

Step 1. Given Information

Solving the given integrals.

$\int x\sinh x^2\mathrm{d}x$

Step 2. Using the substitution method.

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

Step 3. This substitution changes the integral into 

$$\begin{gathered} \int x\sinh x^2\mathrm{d}x=\frac{1}{2}\int \sinh u\mathrm{d}u \\ \int x\sinh x^2\mathrm{d}x=\frac{1}{2}\cosh u+C \\ \int x\sinh x^2\mathrm{d}x=\frac{1}{2}\cosh x^2+C \end{gathered}$$