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Chapter 1: Problem 1

Evaluate the following integrals: $$ \int \frac{x^{2} d x}{\left(4-x^{2}\right)^{5 / 2}} $$

Short Answer

Expert verified
Question: Determine the integral of the function given by: $$\int \frac{x^{2} \, dx}{\left(4-x^{2}\right)^{5 / 2}}$$ Answer: The integral is given by: $$\int \frac{x^{2} \, dx}{\left(4-x^{2}\right)^{5 / 2}} = \frac{4}{3}(4-x^2)^{3/2} + (4-x^2)^{1/2} + C$$

Step by step solution

01

Perform substitution

Let \(u = 4 - x^2\). Now, differentiate \(u\) with respect to \(x\): \(\frac{du}{dx} = -2x\). So, \(x dx = -\frac{1}{2} du\). Notice that we have \(x^2 dx\) in the integral, so we will write \(x dx\) as \(-\frac{1}{2} du\) and replace \(x^2\) by \(4-u\). Now substitute into the integral: $$\int \frac{x^{2} \, dx}{\left(4-x^{2}\right)^{5 / 2}} = \int \frac{(4-u) \cdot -\frac{1}{2} \, du}{\left(u\right)^{5 / 2}}$$
02

Simplify the integral

Now, simplify the integral expression: $$\int \frac{(4-u) \cdot -\frac{1}{2} \, du}{\left(u\right)^{5 / 2}} = -\frac{1}{2} \int \frac{(4-u)}{u^{5/2}} \, du$$
03

Split the integral

We can split the integral into two separate integrals for easier evaluation: $$-\frac{1}{2} \int \frac{(4-u)}{u^{5/2}} \, du = -\frac{1}{2} \left(\int \frac{4}{u^{5/2}} \, du - \int \frac{u}{u^{5/2}}\, du\right)$$
04

Simplify and integrate

Now, simplify the exponents and integrate each term: $$-\frac{1}{2} \left(\int 4u^{-5/2} \, du - \int u^{-3/2} \, du\right)$$ Integrating each term: $$-\frac{1}{2} \left(\frac{-8}{3}u^{3/2} - 2u^{1/2} + C\right)$$
05

Replace \(u\) and simplify

Now, replace \(u\) with the original expression, \(4 - x^2\), and simplify the result: $$-\frac{1}{2} \left(\frac{-8}{3}(4-x^2)^{3/2} - 2(4-x^2)^{1/2} + C\right)$$ So, the final result is: $$\int \frac{x^{2} \, dx}{\left(4-x^{2}\right)^{5 / 2}} = \frac{4}{3}(4-x^2)^{3/2} + (4-x^2)^{1/2} + C$$

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