Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int(\sqrt[3]{x^{2}}+\sqrt{x^{3}}) d x$$

Short answer

Question: Find the indefinite integral of the function \(\sqrt[3]{x^2} + \sqrt{x^3}\). Answer: \(\int(\sqrt[3]{x^{2}}+\sqrt{x^{3}}) dx = \frac{3}{5}x^{\frac{5}{3}} + \frac{2}{5}x^{\frac{5}{2}} + C\)

Step-by-step solution

Integrate the first term \(\sqrt[3]{x^2}\)

To find the integral of \(\sqrt[3]{x^2}\), we'll rewrite it in exponential form as \(x^{\frac{2}{3}}\). Now we can apply the power rule of integration: $$\int x^{\frac{2}{3}} dx$$ $$= x^{\frac{2}{3} + 1} \cdot \frac{1}{\frac{2}{3} + 1} + C$$ $$= x^{\frac{5}{3}} \cdot \frac{3}{5} + C$$ $$= \frac{3}{5}x^{\frac{5}{3}} + C$$

Integrate the second term \(\sqrt{x^3}\)

To find the integral of \(\sqrt{x^3}\), we'll rewrite it in exponential form as \(x^{\frac{3}{2}}\). Now we can apply the power rule of integration: $$\int x^{\frac{3}{2}} dx$$ $$= x^{\frac{3}{2} + 1} \cdot \frac{1}{\frac{3}{2} + 1} + C$$ $$= x^{\frac{5}{2}} \cdot \frac{2}{5} + C$$ $$= \frac{2}{5}x^{\frac{5}{2}} + C$$

Combine the integrals

Now, we can combine the integrals of the individual terms to get the integral of the entire expression: $$\int(\sqrt[3]{x^{2}}+\sqrt{x^{3}}) dx$$ $$= \frac{3}{5}x^{\frac{5}{3}} + \frac{2}{5}x^{\frac{5}{2}} + C$$

Differentiate the result

Now, we'll differentiate the resulting function to verify that it yields the original expression. Differentiating each term separately: $$\frac{d}{dx} \left( \frac{3}{5}x^{\frac{5}{3}} + \frac{2}{5}x^{\frac{5}{2}} \right)$$ $$= \frac{3}{5}\cdot\frac{5}{3}x^{\frac{5}{3} - 1} + \frac{2}{5}\cdot\frac{5}{2}x^{\frac{5}{2}-1}$$ $$= x^{\frac{2}{3}} + x^{\frac{3}{2}}$$ Our differentiation gave us the original expression, so our integration was correct: $$\int(\sqrt[3]{x^{2}}+\sqrt{x^{3}}) dx = \frac{3}{5}x^{\frac{5}{3}} + \frac{2}{5}x^{\frac{5}{2}} + C$$