Q. 50 - 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: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

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

Step-by-Step Solution

Answer

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

Step 1. Given Information.

Given an integral$\int ({\sec }^2\left(x\right) + {\csc }^2\left(x\right))dx.$

Step 2. Formula involved.

$$\begin{gathered} \int {\sec }^2\left(x\right)dx = \tan \left(x\right), and \\ \int cs{c}^2\left(x\right)dx = -cot\left(x\right). \end{gathered}$$

Step 3. Solving the integral.

$$\begin{gathered} \int ({\sec }^2\left(x\right) + {\csc }^2\left(x\right))dx \\ = \int {\sec }^2\left(x\right)dx+ \int {\csc }^2\left(x\right)dx. \\ = \tan \left(x\right) - cot\left(x\right) +c. \end{gathered}$$