Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$).

localid="1648642820410" $\int \left(\tan h\left(x\right)+x{(\sec h\left(x\right))}^2\right)dx.$

Short answer

The value of the given integral is$x\tan h\left(x\right)+c.$

Step-by-step solution

Step 1. Given Information.

Given is a integral:$\int \left(\tan h\left(x\right)+x{(\sec h\left(x\right))}^2\right)dx.$

Step 2. Formula involved.

$$\begin{gathered} uv=\int udv+\int vdu, \\ u\int dv=uv-\int vdu. \end{gathered}$$

Step 3. Solving the integral.

$$\begin{gathered} \int \left(\tan h\left(x\right)+xsec{h}^2\left(x\right)\right)dx \\ =\int \tan h\left(x\right)dx+\int xsec{h}^2\left(x\right)dx \\ Fromstep2weget\int xsec{h}^2\left(x\right)dx=x\tan h\left(x\right)-\int \tan h\left(x\right)dx \\ =\int \tan h\left(x\right)dx+x\tan h\left(x\right)-\int \tan h\left(x\right)dx \\ =x\tan h\left(x\right)+c. \end{gathered}$$