/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 42

Evaluate the following integrals. $$\int \frac{d x}{x^{3} \sqrt{x^{2}-100}}, x>10$$

Short Answer

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Question: Evaluate the integral $$\int \frac{d x}{x^{3} \sqrt{x^{2}-100}}, x>10.$$ Answer: $$-\frac{(x^2 - 100)^{\frac{1}{2}}}{2x^2} + C$$

Step by step solution

01

Identify the substitution

Let's make the following substitution: $$u = x^2 - 100.$$ We will need to find the derivative of this expression with respect to \(x\) to replace \(dx\) in the integral.
02

Find the derivative of the substitution

Differentiating \(u\) with respect to \(x\), we get: $$\frac{du}{dx} = 2x.$$ Now, we need to rearrange it for \(dx\): $$dx = \frac{du}{2x}.$$
03

Substitute into the integral

Now, substitute \(u\) and \(dx\) into the original integral, $$\int \frac{d x}{x^{3} \sqrt{x^{2}-100}}, x>10.$$ We get: $$\int \frac{1}{x^3 \sqrt{u}} \cdot \frac{du}{2x}.$$ Simplify the integral: $$\frac{1}{2} \int \frac{du}{x^4 \sqrt{u}}.$$
04

Rewrite the integral in terms of u

We need to express the whole integral in terms of \(u\). To do this, utilize the original substitution \(u = x^2 - 100\): $$x^2 = u + 100$$ Now, substitute this back into the integral: $$\frac{1}{2} \int \frac{du}{(u+100)^2\sqrt{u}}.$$
05

Evaluate the integral using the power rule

To evaluate this integral, we follow the power rule: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ for all values n not equal to -1. Rewrite the integral as follows: $$\frac{1}{2} \int u^{-\frac{1}{2}}(u+100)^{-2} du$$ Now, use the power rule to integrate this expression: $$-u^{\frac{1}{2}} (u + 100)^{-1}/2 + C$$
06

Substitute back x

Finally, substitute the expression for \(u\) back in terms of \(x\): $$-\frac{(x^2 - 100)^{\frac{1}{2}}}{2(x^2 - 100 + 100)} + C$$ Simplify the expression: $$-\frac{(x^2 - 100)^{\frac{1}{2}}}{2x^2} + C$$ The evaluated integral is thus: $$-\frac{(x^2 - 100)^{\frac{1}{2}}}{2x^2} + C$$

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Most popular questions from this chapter

Many methods needed Show that \(\int_{0}^{\infty} \frac{\sqrt{x} \ln x}{(1+x)^{2}} d x=\pi\) in the following steps. a. Integrate by parts with \(u=\sqrt{x} \ln x.\) b. Change variables by letting \(y=1 / x.\) c. Show that \(\int_{0}^{1} \frac{\ln x}{\sqrt{x}(1+x)} d x=-\int_{1}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} d x\) and conclude that \(\int_{0}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} d x=0.\) d. Evaluate the remaining integral using the change of variables \(z=\sqrt{x}\) (Source: Mathematics Magazine 59, No. 1 (February 1986): 49).

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