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Chapter 6: Problem 21

Use integration tables to find the integral. $$ \int \frac{x^{3}}{\sqrt{4-x^{2}}} d x $$

Short Answer

Expert verified
The integral of \(\int \frac{x^{3}}{\sqrt{4-x^{2}}} d x = -1/2 (8\sqrt{4-x^2} - 2/3 (4-x^2)^{3/2} + C)\).

Step by step solution

01

Substitution

Start the solution by choosing \(u = 4 - x^2\). Then find the differential of \(u\), which is \(du = -2x dx\). Rewrite the integral in terms of \(u\), and for that divide both sides of \(du = -2x dx\) by \(-2x\), to obtain \( dx = -du/2x\). Substitute \(x^2\) from \(u = 4 - x^2\) to be \(x^2 = 4 - u\). Now replace \(dx\) and \(x\) in the integral to rewrite it in terms of \(u\). The integral becomes \(\int \frac{(4-u)(-du/2\sqrt{u}}{\sqrt{u}}\).
02

Simplification

Simplify the integral after substitution. Degree of \(u\) in numerator and denominator will cancel out, so simplify it to \(-1/2 \int \frac{4-u}{\sqrt{u}} du\). This simplifies further to \(-1/2 \int (4u^{-1/2} - u^{1/2}) du\) by distributing the numerator across the denominator.
03

Using the Power Rule for Integration

Now apply the power rule for integral, which states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), unless \(n = -1\). Use this rule to the each term in the integral \(-1/2 \int (4u^{-1/2} - u^{1/2}) du\), and we obtain \(-1/2 (8u^{1/2} - 2/3 u^{3/2} + C)\).
04

Replace \(u\) with original variable

Replace \(u\) with the original variable \(x\). Therefore, the final answer is \(-1/2 (8\sqrt{4-x^2} - 2/3 (4-x^2)^{3/2} + C)\).

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Most popular questions from this chapter

(a) The improper integrals \(\int_{1}^{\infty} \frac{1}{x} d x \quad\) and \(\int_{1}^{\infty} \frac{1}{x^{2}} d x\) diverge and converge, respectively. Describe the essential differences between the integrands that cause one integral to converge and the other to diverge. (b) Sketch a graph of the function \(y=\sin x / x\) over the interval \((1, \infty)\). Use your knowledge of the definite integral to make an inference as to whether or not the integral \(\int_{1}^{\infty} \frac{\sin x}{x} d x\) converges. Give reasons for your answer. (c) Use one iteration of integration by parts on the integral in part (b) to determine its divergence or convergence.

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Consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the \(x\) -axis. (c) Find the volume of the solid generated by revolving the region about the \(y\) -axis. $$ y \leq e^{-x}, y \geq 0, x \geq 0 $$

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 2} \sin ^{6} x d x $$

Find the area of the region bounded by the graphs of the equations. $$ y=\cos ^{2} x, \quad y=\sin x \cos x, \quad x=-\pi / 2, \quad x=\pi / 4 $$
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