Q. 63 - Calculus

Question

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

$a\ne 0 , b\ne 0 , c>1, and c\ne 0.$

$\int a{e}^{bx+c}dx.$

Step-by-Step Solution

Answer

The value of the given integral is $\frac{a{e}^{bx+c}}{b}+k, where k is a cons\tan t.$

Step 1. Given Information.

Given is a integral: $\int a{e}^{bx+c}dx ,$

 $$\begin{gathered} a\ne 0 , b\ne 0 , c>1, and c\ne 0, \\ and a,b, and c are cons\tan ts. \end{gathered}$$

Step 2. Formula involved.

$$\begin{gathered} \int e^{bx}dx = \frac{e^{bx}}{b}+k, \\ \int a{e}^{x}dx = a{e}^x+k. \end{gathered}$$

Step 3. Solving the integral.

$$\begin{gathered} \int a{e}^{bx+c}dx \\ = a\int e^{bx+c}dx \\ let t=bx+c \\ dt =bdx \\ Putting in the integral we get, \\ = \frac{a}{b}\int e^{t}dt \\ = \frac{a}{b}{e}^t +k \\ = \frac{a}{b}{e}^{bx+c} +k. \end{gathered}$$