Q. 52 - 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: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

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

Step-by-Step Solution

Answer

The value of the integral is $x\tan x+c.$

Step 1. Given Information.

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

Step 2. Formula involved.

$uv = \int udv + \int vdu.$

Step 3. Solving the integral.

$$\begin{gathered} \int (\tan x+x{\sec }^2(x\left)\right)dx \\ = \int \tan xdx +\int x{\sec }^2\left(x\right)dx \\ = \int \tan xdx + x\tan x - \int \tan xdx , \\ [Since x\tan x = \int xd(\tan x) + \int \tan xdx] \\ = x\tan x+c. \end{gathered}$$