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Chapter 1: Problem 9

Evaluate the following integrals: $$ \int \frac{x^{2}}{\sqrt{\left(a^{6}-x^{6}\right)}} d x $$

Short Answer

Expert verified
Question: Evaluate the following integral: $$\int \frac{x^2}{\sqrt{a^6 - x^6}} dx$$ Answer: $$-\frac{3}{2a^3} \sqrt{a^6 - x^6} + C$$

Step by step solution

01

Choose a substitution

We are going to make the substitution: $$ u = x^3 \implies du = 3x^2 dx $$ We will also change the \(x^2\) in the numerator to make everything in terms of \(u\): $$ x^2 = u^{2/3} $$ Now, we rewrite the integral in terms of \(u\).
02

Rewrite the integral in terms of u

Rewriting the integral with our substitution: $$ \int \frac{u^{2/3}}{\sqrt{a^6 - u^2}} (3u^{1/3}) du $$ We can simplify this expression further.
03

Simplify the integral

We can simplify the integral as follows: $$ 3 \int \frac{u}{\sqrt{a^6 - u^2}} du $$
04

Evaluate the integral

Now, let's evaluate the integral: $$ 3 \int \frac{u}{\sqrt{a^6 - u^2}} du = -\frac{3}{2a^3} \sqrt{a^6 - u^2} + C $$ The function is now in the reduced form, but it is still in terms of \(u\). Our final step is to convert it back to \(x\).
05

Write the result in terms of x

Now we substitute back the value of \(u = x^3\), and the result will be in terms of \(x\): $$ -\frac{3}{2a^3} \sqrt{a^6 - x^6} + C $$ That's our final answer for the given integral.

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Most popular questions from this chapter

Evaluate the following integrals : $$ \int \frac{\left(1-x^{2}\right) d x}{x^{1 / 2}\left(1+x^{2}\right)^{3 / 2}} $$

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