/*! 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 33 Find the indefinite integral and... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 33

Find the indefinite integral and check your result by differentiation. $$ \int \sqrt[3]{x^{2}} d x $$

Short Answer

Expert verified
The indefinite integral of \( \sqrt[3]{x^{2}} \) is \( \frac{3}{5}x^{5/3} + C \). Checking the result by differentiating, we obtained the original function \( \sqrt[3]{x^{2}} \), implying that the integral was calculated correctly.

Step by step solution

01

Identify the function to be integrated

The function given that needs to be integrated is \( \sqrt[3]{x^{2}} \)
02

Convert the root to an exponent

It is more simple to handle an exponent rather than a root. The fraction 2/3 will be used in place of the cubic root, resulting in \(x^{2/3}\)
03

Apply integral rules

According to the power rule for integration, which states that the integral of \(x^{n}\) is \(x^{n+1}/(n+1)\), apply this rule to \(x^{2/3}\) to get the integral.
04

Compute the Integral

Integrating \(x^{2/3}\), you obtain \( \frac{3}{5}x^{5/3} + C\), where C is the constant of integration.
05

Differentiate to Check

Next, differentiate the result \(\frac{3}{5}x^{5/3} + C\) back to check if it gives the integrand \(x^{2/3}\). According to the power rule of differentiation, the derivative \( \frac{5}{3}*\frac{3}{5}x^{(5/3-1)} \) becomes \(x^{2/3}\). This confirms that the calculation of the integral is correct.

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