Q. 64 - Calculus

Question

Solve each of the integrals in Exercises 63–68, where a, b, and c are real numbers with $a\ne 0 , b\ne 0 , c>1, and c\ne 0.$

$\int \frac{a}{b{x}^c}dx.$

Step-by-Step Solution

Answer

The value of the given integral is: $\frac{a{x}^{1-c}}{b(1-c)} + k.$

Step 1. Given Information.

Given is the integral:

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

Step 2. Formula used.

$\int x^{n}dx = \frac{x^{n+1}}{n+1}+k , where n\ne 1.$

Step 3. Solving the integral.

$$\begin{gathered} \int \frac{a}{b{x}^c}dx \\ = \frac{a}{b}\int x^{-c}dx \\ = \frac{a}{b}\frac{x^{-c+1}}{-c+1} + k \\ = \frac{a{x}^{1-c}}{b(1-c)} + k. \end{gathered}$$