/*! 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

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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.

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