Q. 61 - Calculus

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: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

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

Step-by-Step Solution

Answer

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

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.

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