Q. 65 - Calculus

Question

Solve each of the integrals in Exercises 63–68, where a, b, and c are real numbers with 

$$\begin{gathered} a\ne 0 , b\ne 0 , c>1, and c\ne 0. \\ \int \frac{\sec \left(ax\right)\tan \left(ax\right)}{b}dx. \end{gathered}$$

Step-by-Step Solution

Answer

The value of the given integral is $\frac{1}{ab}\sec \left(ax\right) +c.$

Step 1. Given Infromation

Given is the integral:

$$\begin{gathered} \int \frac{\sec \left(ax\right)\tan \left(ax\right)}{b}dx, \\ a\ne 0 , b\ne 0 , c>1, and c\ne 0, \\ and a,b,and c are cons\tan ts. \end{gathered}$$

Step 2. Formula involved.

$\int sec\left(x\right) \tan \left(x\right) dx = sec \left(x\right) +c.$

Step 3. Solving the integral.

$$\begin{gathered} \int \frac{sec\left(ax\right) \tan \left(ax\right)}{b}dx \\ let t = ax \\ dt = adx \\ Putting the value in the integral \\ = \frac{1}{b}\int \frac{sec\left(t\right)\tan \left(t\right) dt}{a} \\ =\frac{1}{ab}\int sec\left(t\right)\tan \left(t\right)dt \\ =\frac{1}{ab}sect +c \\ =\frac{1}{ab}sec\left(ax\right) +c. \end{gathered}$$