/*! 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 9 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 9

Evaluate the following integrals : $$ \int x^{1 / 4}\left(2+3 x^{2}\right)^{3} d x $$

Short Answer

Expert verified
Answer: The best substitution to simplify the integration process for the given integral is \(u = 2 + 3x^2\).

Step by step solution

01

Identify the best substitution

Notice that the integrand has the form \(x^{\frac{1}{4}} (2 + 3x^2)^3\). Let's define the substitution: $$ u = 2 + 3x^2 $$ Now, we will compute the derivative of \(u\) with respect to \(x\) and express it as \(d(u)\) in terms of \(d(x)\).
02

Compute the derivative of the substitution

Differentiating \(u\) with respect to \(x\), we obtain: $$ \frac{du}{dx} = 6x $$ Now, express \(du\) in terms of \(dx\): $$ du = 6x dx $$
03

Make the substitution and simplify the integral

Since we have \(u=2+3x^2\), we can express \(x^{1/4}\) in terms of \(u\) as follows: \(x^{1/4} = (u-2)^{1/4}\). Now, we'll make the substitution in the integral and simplify: $$ \int x^{\frac{1}{4}}(2+3x^2)^3 dx = \int (u-2)^{\frac{1}{4}} u^3 \left(\frac{1}{6} du\right) $$ Now, simplify the integral: $$ \int (u-2)^{\frac{1}{4}} u^3 \left(\frac{1}{6}du\right) = \frac{1}{6} \int (u-2)^{\frac{1}{4}} u^3 du $$
04

Calculate the integral

Now that we have simplified the integral, we can proceed and calculate it: $$ \frac{1}{6}\int (u-2)^{\frac{1}{4}}u^3du $$ To solve this integral, we can use integration by parts or lookup in a table of integrals. The antiderivative of \((u-2)^{\frac{1}{4}}u^3\) is: $$ \frac{1}{6} \left(\frac{4}{7}u^{\frac{7}{4}}(u-2)^{\frac{1}{4}} - \frac{16}{11}u^{\frac{11}{4}}(u-2)^{-\frac{3}{4}} + C\right) $$ where C is the constant of integration.
05

Substitute back in terms of x

Now that we have the antiderivative in terms of \(u\), we need to substitute back in terms of \(x\) using \(u=2+3x^2\): $$ \frac{1}{6} \left(\frac{4}{7}(2+3x^2)^{\frac{7}{4}}(2+3x^2 - 2)^{\frac{1}{4}} - \frac{16}{11}(2+3x^2)^{\frac{11}{4}}((2+3x^2) - 2)^{-\frac{3}{4}} + C\right) $$ Now, simplify the expression: $$ \frac{1}{6} \left(\frac{4}{7}(2+3x^2)^{\frac{7}{4}}(3x^2)^{\frac{1}{4}} - \frac{16}{11}(2+3x^2)^{\frac{11}{4}}((3x^2)^{-\frac{3}{4}} + C\right) $$ This is the final expression for the evaluated integral.

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Most popular questions from this chapter

Evaluate \(\int \frac{9 x^{3}-3 x^{2}+2}{\sqrt{3 x^{2}-2 x+1}} d x\)

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