Joan Thompson

Answered

2021-12-31

Find definite integral.

${\int}_{0}^{2}{x}^{2}{(3{x}^{3}+1)}^{\frac{1}{3}}dx$

Answer & Explanation

Jenny Sheppard

Expert

2022-01-01Added 35 answers

Step 1

The given integral can be solved by the method of substitution. The substitution that will be used is

$3{x}^{3}+1=u$ . This gives $9{x}^{2}dx=du$ or $x}^{2}dx=\frac{du}{9$ .

This substitution absorbs the${x}^{2}dx$ term into $\frac{du}{9}$ . Calculate the corresponding limits of the integration in terms of the new variable u.

Step 2

Limits of integration for x are from 0 to 2. New variable for integration is$u=3{x}^{3}+1$ . So the lower limit of integration in terms of new variable will be 1. Calculate the upper limit by substituting x=2.

$u=3\cdot {2}^{3}+1$

=3*8+1

=25

So, the integration with this substitution becomes$\int}_{1}^{25}{u}^{\frac{1}{3}}\frac{du}{9$ . Calculate this integral using the integral

$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}$ .

$\int}_{1}^{25}{u}^{\frac{1}{3}}\frac{du}{9}=\frac{1}{9}{\left(\frac{{u}^{\frac{1}{3}+1}}{\frac{1}{3}+1}\right)}_{1}^{25$

$=\frac{1}{9}\cdot \frac{3}{4}{\left({u}^{\frac{4}{3}}\right)}_{1}^{25}$

$=\frac{1}{12}({25}^{\frac{4}{3}}-1)$

Hence, the given definite integral is equal to$\frac{1}{12}({25}^{\frac{4}{3}}-1)$

The given integral can be solved by the method of substitution. The substitution that will be used is

This substitution absorbs the

Step 2

Limits of integration for x are from 0 to 2. New variable for integration is

=3*8+1

=25

So, the integration with this substitution becomes

Hence, the given definite integral is equal to

Shawn Kim

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