Question

Solve given definite integral.

${\int }_1^2 \frac{3}{{\left(9-2x^2\right)}^{3/2}}dx$

Short answer

$\frac{2}{3}-\frac{\sqrt{7}}{21}$

Step-by-step solution

Step1. Given Information

The integral is as follows.

${\int }_1^2 \frac{3}{{\left(9-2x^2\right)}^{\frac{3}{2}}}dx$

The objective is to solve the integral.

Step2. Assumptions

To solve the integral, let $u=x^2$

Now, derivation of above equation is solved below.

$$\begin{gathered} u=x^2 \\ du=2xdx \end{gathered}$$

Step3. Solution

The integral is solved below.

$\begin{array}{r}\int \frac{3}{{\left(9-2x^2\right)}^{\frac{3}{2}}}dx\\ =\frac{x}{3\sqrt{9-2x^2}}\end{array}$

Step4. Solution

The definite integral is solved below.

${\int }_1^2 \frac{3}{{\left(9-2x^2\right)}^{\frac{3}{2}}}dx$

$={\left[\frac{x}{3\sqrt{9-2x^2}}\right]}_1^2$

$=\left[\frac{2}{3\sqrt{9-2(2{)}^2}}\right]-\left[\frac{1}{3\sqrt{9-2(1{)}^2}}\right]$

$=\frac{2}{3}-\frac{\sqrt{7}}{21}$

Therefore, the value is$\frac{2}{3}-\frac{\sqrt{7}}{21}$