Question

evaluate the following indefinite integral.
int 9 x ^ { \frac { 1 } { 2 } } d x
int 9 x ^ { \frac { 1 } { 2 } } d x =

Explanation:

Step1: Use the constant - multiple rule of integration

The constant - multiple rule states that \(\int kf(x)dx = k\int f(x)dx\) where \(k = 9\) and \(f(x)=x^{\frac{1}{2}}\). So, \(\int9x^{\frac{1}{2}}dx=9\int x^{\frac{1}{2}}dx\).

Step2: Use the power rule of integration

The power rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). Here \(n=\frac{1}{2}\). Then \(9\int x^{\frac{1}{2}}dx=9\times\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C\).

Step3: Simplify the expression

First, simplify \(\frac{1}{2}+1=\frac{1 + 2}{2}=\frac{3}{2}\). Then \(9\times\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+C\). Since \(\frac{9}{\frac{3}{2}}=9\times\frac{2}{3}=6\), the integral becomes \(6x^{\frac{3}{2}}+C\).

Answer:

\(6x^{\frac{3}{2}}+C\)