Chapter 7, Problem 1RE

Evaluate the given integral with the aid of an appropriate u-substitution.

${\displaystyle \int \sqrt{4+9x}}dx$

To determine

To calculate: The solution of the given integral with the aid of an appropriate $u\text{-substitution}$ .

${\displaystyle \int \sqrt{4+9x}}dx$

Answer to Problem 1RE

The $u\text{-substitution}$ integral is $\frac{2}{27}{\left(4+9x\right)}^{3/2}+C$ .

Explanation of Solution

Calculation:

Consider the given equation.

${\displaystyle \int \sqrt{4+9x}}dx$

Let $u=4+9x$ .

Then,

${\displaystyle \int \sqrt{4+9x}}dx={\displaystyle \int \sqrt{u}}dx$

Then differentiate both side of the equation $u=4+9x$ .

$\begin{array}{c}9dx=du\\ dx=\frac{1}{9}du\end{array}$

Now substitute $dx=\frac{1}{9}du$ in the integral ${\displaystyle \int \sqrt{u}}dx$ and solve as,

$\begin{array}{c}{\displaystyle \int \sqrt{u}}dx=\frac{1}{9}{\displaystyle \int \sqrt{u}}du\\ =\frac{1}{9}{\displaystyle \int u^{1/2}}du\\ =\frac{1}{9}\left(\frac{u^{1/2+1}}{1/2+1}\right)+C\\ =\frac{1}{9}\left(\frac{u^{3/2}}{3/2}\right)+C\end{array}$

Further solve as,

${\displaystyle \int \sqrt{u}}dx=\frac{2}{27}{u}^{3/2}+C$

Now, substitute $u=4+9x$ in the above equation,

${\displaystyle \int \sqrt{4+9x}}dx=\frac{2}{27}{\left(4+9x\right)}^{3/2}+C$

Therefore, the $u\text{-substitution}$ integral is $\frac{2}{27}{\left(4+9x\right)}^{3/2}+C$ .