Question

find the indefinite integral.

Answer 1

Using substitution, the result of the indefinite integral is:

$3\ln{(1 + x^{\frac{1}{3}})} + C$

What is the integral?

The integral is:

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

The substitution is:

$u = 1 + x^{\frac{1}{3}}$

Then the derivative is:

$du = \frac{1}{3}x^{-\frac{2}{3}} dx$

ttex]dx = 3dux^{-\frac{2}{3}}[/tex]

Hence the integral is:

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

$\int \frac{1}{x^{\frac{2}{3}}u} \times 3dux^{-\frac{2}{3}}$

$\int \frac{3du}{u}$

$3\ln{u} + C$

Returning to x:

$3\ln{(1 + x^{\frac{1}{3}})} + C$

More can be learned about integration by substitution at https://brainly.com/question/13058734

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