Problem 19

Question

Find the indefinite integral and check your result by differentiation. $$ \int y^{3 / 2} d y $$

Step-by-Step Solution

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Answer

The indefinite integral $\int y^{3 / 2} d y$ is $2/5 y^{5/2} + C$. After differentiating this result, we indeed get back the original function $y^{3 / 2}$.

1Step 1: Compute the Integral

Applying the power rule for integration to $\int y^{3/2}dy$, we have the integral as $1/(3/2+1) y^{3/2+1} + C = 2/5 y^{5/2} + C $

2Step 2: Check the Result Using Differentiation

Now differentiate the result to check if the differentiated function matches the integrand. Differentiating $2/5 y^{5/2} + C$ using the power rule yields $1/2 * 5/2 y^{5/2-1} = y^{3/2}$. This matches the original integrand, so the integral is correct

3Step 3: Confirm the Answer

The indefinite integral of $y^{3/2}$ is $2/5 y^{5/2} + C$ and the differentiation of the integral gives back the original function $y^{3/2}$.

Problem 19

Problem 20

Other exercises in this chapter

Problem 19

Use the Log Rule to find the indefinite integral. $$ \int \frac{x}{x^{2}+1} d x $$

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Problem 19

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x+1}{\left(x^{2}+2 x-3\right)^{2}} d x $$

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Problem 20

Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}}{3-x^{3}} d x $$

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Problem 20

Find the indefinite integral and check the result by differentiation. $$ \int \frac{6 x}{\left(1+x^{2}\right)^{3}} d x $$

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