Identify Integral: Identify the integral that needs to be solved. We need to solve the integral of the cube root of 5 times x squared, which is written as ∫(5^(1/3) * x^2) dx.Rewrite Integral: Rewrite the integral in a more convenient form. We can rewrite the integral as 5^(1/3) * ∫x^2 dx, since 5^(1/3) is a constant and can be factored out of the integral.Apply Power Rule: Apply the power rule for integration. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. So, ∫x^2 dx = (x^(2+1))/(2+1) + C = (x^3)/3 + C.Combine with Constant: Combine the constant with the antiderivative. Now we multiply the antiderivative by the constant factor we factored out earlier, which gives us (5^(1/3) * (x^3)/3) + C.Write Final Answer: Write the final answer. The final answer is (5^(1/3) * (x^3)/3) + C.

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$\int 3\sqrt[5]{x^{2}}\,dx$

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\int 3\sqrt[5]{x^{2}}\,dx

Identify Integral: Identify the integral that needs to be solved. $\newline$ We need to solve the integral of the cube root of $5$ times $x$ squared, which is written as $\int(5^{1/3} \cdot x^2) \, dx$ .
Rewrite Integral: Rewrite the integral in a more convenient form. $\newline$ We can rewrite the integral as $5^{\frac{1}{3}} \times \int x^2 \, dx$ , since $5^{\frac{1}{3}}$ is a constant and can be factored out of the integral.
Apply Power Rule: Apply the power rule for integration. $\newline$ The power rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $C$ is the constant of integration. $\newline$ So, $\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$ .
Combine with Constant: Combine the constant with the antiderivative. $\newline$ Now we multiply the antiderivative by the constant factor we factored out earlier, which gives us $5^{\frac{1}{3}} \cdot \frac{x^3}{3}$ + $C$ .
Write Final Answer: Write the final answer. $\newline$ The final answer is $5^{\frac{1}{3}} \cdot \frac{x^3}{3} + C$ .