Q. 66 - Calculus

Question

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

$$\begin{gathered} a\ne 0 , b\ne 0 , c>1, and c\ne 0. \\ \int \left(b\left(c^x\right)+a\right)dx. \end{gathered}$$

Step-by-Step Solution

Answer

The value of the integral is $b{c}^x\ln c + ax +k.$

Step 1. Given Information.

Given is a integral:

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

Step 2. Formula involved.

$\int a^{x}dx = a^x\ln a +k.$

Step 3. Solving the integral.

$$\begin{gathered} \int \left(b\right(c^x)+a)dx \\ = \int b\left(c^x\right)dx +\int adx \\ = b{c}^x\ln c +ax +k. \end{gathered}$$