# Evaluate the integral: { x^2 \sqrt{( 9 - x^2 )} } using u-substitution, and not trigonometric...

## Question:

Evaluate the integral {eq}\int x^2 \sqrt{( 9 - x^2 )} \, dx {/eq} using {eq}u {/eq}-substitution, and not trigonometric substitution.

## Integration:

If we want to get the function back then, we integrate derivative. The opposite of the derivative known as integration. This rule to get back function is known as the antiderivative. The integration sum of the area of the sub rectangles within the subintervals of the limit. The formula we use here shown below:

{eq}\int \sqrt{a^{2}-x^{2}}dx=\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}\sin ^{-1}(\frac{x}{a})+C {/eq}... (1)

Given a function {eq}\int x^2 \sqrt{( 9 - x^2 )} \, dx {/eq}.

{eq}\begin{align*} \int x^2 \sqrt{( 9 - x^2 )}dx&=\int \sqrt{( 9x^{4} - x^{6} )}dx\\ &=\int x\sqrt{( 9x^{2} - (x^{2})^{2} )}dx\\ \end{align*} {/eq}.

Let {eq}x^{2}=t {/eq} and differentiate for {eq}x {/eq} then, {eq}xdx=\frac{1}{2}dt {/eq}.

{eq}\begin{align*} \int x^2 \sqrt{( 9 - x^2 )}dx&=\frac{1}{2}\left [ \frac{( x^{2}-\frac{9}{2})}{2} \sqrt{( 9 - x^2 )}+\frac{81}{8}\sin ^{-1}(\frac{ x^{2}-\frac{9}{2}}{\frac{9}{2}})\right ]+C\\ &=\frac{1}{2}\left [\frac{\frac{( 2x^{2}-9)}{2}}{2} \sqrt{( 9 - x^2 )}+\frac{81}{8}\sin ^{-1}(\frac{ 2x^{2}-9}{9})\right] +C\\ &=\frac{1}{2}\left [ \frac{( 2x^{2}-9)}{4} \sqrt{( 9 - x^2 )}+\frac{81}{8}\sin ^{-1}(\frac{ 2x^{2}-9}{9})\right ]+C\\ \end{align*} {/eq}.

Put the value of {eq}t {/eq} in terms of {eq}x {/eq}.

{eq}\begin{align*} \int x^2 \sqrt{( 9 - x^2 )}dx&=\frac{1}{2}\left [ \frac{( x^{2}-\frac{9}{2})}{2} \sqrt{( 9 - x^2 )}+\frac{81}{8}\sin ^{-1}(\frac{ x^{2}-\frac{9}{2}}{\frac{9}{2}})\right ]+C\\ &=\frac{1}{2}\left [\frac{\frac{( 2x^{2}-9)}{2}}{2} \sqrt{( 9 - x^2 )}+\frac{81}{8}\sin ^{-1}(\frac{ 2x^{2}-9}{9})\right] +C\\ &=\frac{1}{2}\left [ \frac{( 2x^{2}-9)}{4} \sqrt{( 9 - x^2 )}+\frac{81}{8}\sin ^{-1}(\frac{ 2x^{2}-9}{9})\right ]+C\\ \end{align*} {/eq}.

Thus, the integral is {eq}\frac{1}{2}\left [ \frac{( 2x^{2}-9)}{4} \sqrt{( 9 - x^2 )}+\frac{81}{8}\sin ^{-1}(\frac{ 2x^{2}-9}{9})\right ]+C {/eq}.