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Chapter 6: Problem 21

Find the integral. $$ \int \frac{\sqrt{1-x^{2}}}{x^{4}} d x $$

Short Answer

Expert verified
The integral of the given function is \( -|x| - 1 \) .

Step by step solution

01

Apply Substitution

A good choice for a substitution is \(u = x^{2}\), so \( du = 2xdx \). Also the integral changes to \( \frac{1}{2} \int \frac{\sqrt {1 - u}}{u^{2}} du \).
02

Simplify the integral

Reorganize the integral function into a form comfortable for resolving. The integral now becomes \( -\frac{1}{2} \int \frac{u'^{1/2}}{u} du \).
03

Apply Power rule for integration

Now we can easily find the integral using a standard rule. This requires recognizing one part of the function as the derivative of another part. The integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). The integral thus becomes: \( -\frac{1}{2}(2\sqrt{u} + 2) = -\sqrt{u} - 1 \).
04

Substitute back u

Now we must revert to the original variable, x. Replacing u by \(x^{2}\), we find: The integral equals \( -\sqrt{x^{2}} - 1 = -|x| - 1\).

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of \(f\) is symmetric with respect to the origin or the \(y\) -axis, then \(\int_{0}^{\infty} f(x) d x\) converges if and only if \(\int_{-\infty}^{\infty} f(x) d x\) converges

(A) find the indefinite integral in two different ways. (B) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (C) Verify analytically that the results differ only by a constant. $$ \int \sec ^{4} 3 x \tan ^{3} 3 x d x $$

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\sin a t $$

Describe the different types of improper integrals

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then \(f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t\).
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