/*! 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 27 Find the integral. $$ \int \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 27

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

Short Answer

Expert verified
\(-\frac{1}{x^{2}+2} + C\)

Step by step solution

01

Substitution

Let \( u = x^{2}+2 \), then \( du = 2xdx \) and \( xdx = \frac{1}{2} du \). The equation \(\int \frac{1}{4+4 x^{2}+x^{4}} d x\) will then become \(\int \frac{1}{2u^{2}} du\)
02

Apply the power rule for integrals

The power rule states that \(\int u^{-n} du = \frac{1}{1-n}u^{1-n} + C\), where \(C\) is the constant of integration. Now apply this rule to our integral, with \(n=2\) we get \( \int u^{-2} du = \frac{1}{1-2}u^{1-2} + C = -u^{-1} + C\)
03

Substitute back

Now substitute the substitution from step 1 back in: \(-u^{-1} + C = -\frac{1}{u} + C = -\frac{1}{x^{2}+2} + C\)
04

Final answer

To conclude, the integral \(\int \frac{1}{4+4 x^{2}+x^{4}} d x\) equals to \(-\frac{1}{x^{2}+2} + C\) where \(C\) is the constant of integration.

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

In Exercises 27-30, verify the integration formula. $$ \int \frac{u^{2}}{(a+b u)^{2}} d u=\frac{1}{b^{3}}\left(b u-\frac{a^{2}}{a+b u}-2 a \ln |a+b u|\right)+C $$

Fluid Force Find the fluid force on a circular observation window of radius 1 foot in a vertical wall of a large water-filled tank at a fish hatchery when the center of the window is (a) 3 feet and (b) \(d\) feet \((d>1)\) below the water's surface. Use trigonometric substitution to evaluate the one integral.

Given continuous functions \(f\) and \(g\) such that \(0 \leq f(x) \leq g(x)\) on the interval \([a, \infty),\) prove the following. (a) If \(\int_{a}^{\infty} g(x) d x\) converges, then \(\int_{a}^{\infty} f(x) d x\) converges. (b) If \(\int_{a}^{\infty} f(x) d x\) diverges, then \(\int_{a}^{\infty} g(x) d x\) diverges.

For the region bounded by the graphs of the equations, find: (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. $$ y=\sin x, y=0, x=0, x=\pi $$

In Exercises \(41-44,\) find or evaluate the integral using substitution first, then using integration by parts. $$ \int \sin \sqrt{x} d x $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.