/*! 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 41 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 41

Evaluate the following integrals. $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} dx$$ Answer: The evaluated integral is $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} dx =-\frac{1}{10}x\sqrt{100-x^2} + 5\arcsin\frac{x}{10} + C$$

Step by step solution

01

Trigonometric Substitution

To simplify the integral, perform the trigonometric substitution \(x = 10\sin{\theta}\), which also implies \(dx = 10\cos{\theta}d\theta\).
02

Substitute and Simplify

Replace \(x\) and \(dx\) in the integral with the substitution: $$\int \frac{(10\sin{\theta})^2}{(100 - (10\sin{\theta})^2)^{3 / 2}} (10\cos{\theta}d\theta) = \int \frac{100\sin^2{\theta}}{(100(1 - \sin^2{\theta}))^{3 / 2}} (10\cos{\theta}d\theta)$$ Using the identity \(1 - \sin^2{\theta} = \cos^2{\theta}\), the integral becomes: $$\int \frac{100\sin^2{\theta}}{(100\cos^2{\theta})^{3 / 2}} (10\cos{\theta}d\theta) = \int \frac{100\sin^2{\theta}}{100\cos^3{\theta}} (10\cos{\theta}d\theta)$$ Simplify the integral further: $$\int 10\sin^2{\theta} d\theta$$
03

Use the Power-Reduction Formula

To solve this integral, use the power-reduction formula \(\sin^2{\theta} = \frac{1 - \cos{2\theta}}{2}\): $$\int 10\sin^2{\theta} d\theta = \int 10\left(\frac{1 - \cos{2\theta}}{2}\right) d\theta$$ Distribute the \(10\): $$\int 10\cdot\frac{1}{2} - 10\cdot\frac{\cos{2\theta}}{2} d\theta = 5\int (1 - \cos{2\theta})d\theta$$
04

Antiderivative of the Integrand

Now, find the antiderivative: $$5\int (1 - \cos{2\theta})d\theta = 5\left(\int d\theta - \int \cos{2\theta} d\theta\right)$$ $$ = 5(\theta - \frac{1}{2}\sin{2\theta}) + C$$
05

Reverse the Trigonometric Substitution

To reverse the trigonometric substitution, recall that we made the substitution \(x = 10\sin{\theta}\). Therefore, \(\sin{\theta} = \frac{x}{10}\). Hence, we need to find \(\theta\) and \(\sin{2\theta}\) from this. Using the Pythagorean identity, \(\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - \frac{x^2}{100}} = \frac{\sqrt{100-x^2}}{10}\). Now, using the double angle identity for sine, \(\sin{2\theta} = 2\sin\theta\cos\theta = 2\frac{x}{10}\frac{\sqrt{100-x^2}}{10} = \frac{x\sqrt{100-x^2}}{50}\).
06

Substitute Back and Simplify

Substitute everything back into the integral: $$5(\theta - \frac{1}{2}\sin{2\theta}) + C = 5\left(\arcsin\frac{x}{10} - \frac{1}{2}\frac{x\sqrt{100-x^2}}{50}\right) + C$$ Finally, simplify the answer: $$=-\frac{1}{10}x\sqrt{100-x^2} + 5\arcsin\frac{x}{10} + C$$ So, the evaluated integral is: $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} d x =-\frac{1}{10}x\sqrt{100-x^2} + 5\arcsin\frac{x}{10} + C$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Decaying oscillations Let \(a>0\) and \(b\) be real numbers. Use integration to confirm the following identities. a. \(\int_{0}^{\infty} e^{-a x} \cos b x d x=\frac{a}{a^{2}+b^{2}}\) b. \(\int_{0}^{\infty} e^{-a x} \sin b x d x=\frac{b}{a^{2}+b^{2}}\)

Consider the curve \(y=\ln x\) a. Find the length of the curve from \(x=1\) to \(x=a\) and call it \(L(a) .\) (Hint: The change of variables \(u=\sqrt{x^{2}+1}\) allows evaluation by partial fractions.) b. Graph \(L(a)\) c. As \(a\) increases, \(L(a)\) increases as what power of \(a ?\)

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{x-\sqrt[4]{x}} ; x=u^{4}$$

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=t \longrightarrow F(s)=\frac{1}{s^{2}}$$

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{2}{x\left(x^{2}+1\right)^{2}} d x$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.