Q. 55 - Calculus

Question

Solve each of the integrals in Exercises 21–66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.) 

$\int {\cos }^3\left(x\right){\sec }^2\left(x\right)dx$

Step-by-Step Solution

Answer

The soluton of the integral is $\sin \left(x\right)+C$.

Step 1. Given Information

The given integral is $\int {\cos }^3\left(x\right){\sec }^2\left(x\right)dx$.

Step 2. Rewrite

• Use the trigonometric identities to rewrite the integral as follows:

$\begin{array}{rcl}\int {\cos }^3\left(x\right){\sec }^2\left(x\right)dx& =& \int \frac{{\cos }^3\left(x\right)}{{\cos }^2\left(x\right)}dx\\ & =& \int \cos \left(x\right)dx\end{array}$

Step 3. Integrate

Integrate the obtained integral using the trigonometric formulas.

$\int \cos \left(x\right)dx=\sin \left(x\right)+C$