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Chapter 4: Problem 34

Determine the following indefinite integrals. Check your work by differentiation. $$\int \sqrt[5]{r^{2}} d r$$

Short Answer

Expert verified
Question: Find the indefinite integral of the function $$\sqrt[5]{r^2}$$. Verify your answer by differentiation. Solution: The indefinite integral of the given function is $$\int \sqrt[5]{r^2} dr = \frac{5r^{\frac{7}{5}}}{7} + C$$. We verify our answer through differentiation and find that $$\frac{d}{dr}\left(\frac{5r^{\frac{7}{5}}}{7} + C\right) = r^{\frac{2}{5}}$$.

Step by step solution

01

Identify the Function and its Power

In this problem, the given function is $$\sqrt[5]{r^2}$$, which can be written as $$r^{\frac{2}{5}}$$. Now we will find its integral.
02

Apply the Integration Formula

Now, we will apply the integration formula for power functions: $$\int r^{\frac{2}{5}} dr = \frac{r^{\frac{2}{5} + 1}}{\frac{2}{5} + 1} + C$$
03

Simplify the Expression

We will simplify the expression: $$\frac{r^{\frac{7}{5}}}{\frac{7}{5}} + C$$ To get rid of the fraction in the denominator, we can multiply the numerator and denominator by 5: $$\frac{5r^{\frac{7}{5}}}{7} + C$$ Thus, the indefinite integral of the given function is: $$\int \sqrt[5]{r^2} dr = \frac{5r^{\frac{7}{5}}}{7} + C$$
04

Check the Result by Differentiation

Now, we will differentiate our result to ensure its accuracy: $$\frac{d}{dr}\left(\frac{5r^{\frac{7}{5}}}{7} + C\right)$$ Using the power rule for differentiation: $$\frac{d}{dr}\left(\frac{5r^{\frac{7}{5}}}{7}\right) = \frac{5}{7} \cdot \frac{7}{5} r^{\frac{7}{5} - 1} = r^{\frac{2}{5}}$$ Since the differentiation of our result matches the given function, our work is verified: $$\int \sqrt[5]{r^2} dr = \frac{5r^{\frac{7}{5}}}{7} + C$$

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