Evaluate the integral. intx^(2)(x+5)^(2)dx Answer:

Expand Integrands: To evaluate the integral of x^2(x+5)^2, we first expand the integrand to make the integration process straightforward. I = ∫x^2(x+5)^2 dx I = ∫x^2(x^2 + 10x + 25) dx I = ∫(x^4 + 10x^3 + 25x^2) dxIntegrate Each Term: Now we integrate each term separately. I = ∫x^4 dx + ∫10x^3 dx + ∫25x^2 dx I = (1/5)x^5 + (10/4)x^4 + (25/3)x^3 + C I = (1/5)x^5 + (5/2)x^4 + (25/3)x^3 + CSimplify Final Answer: We simplify the expression to get the final answer. I = (1/5)x^5 + (5/2)x^4 + (25/3)x^3 + C

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Calculus

Find indefinite integrals using the substitution and by parts

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Q. Evaluate the integral.

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\int x^{2}(x+5)^{2} \mathrm{~d} x

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Answer:

Expand Integrands: To evaluate the integral of $x^2(x+5)^2$ , we first expand the integrand to make the integration process straightforward. $\newline$ $I = \int x^2(x+5)^2 \, dx$ $\newline$ $I = \int x^2(x^2 + 10x + 25) \, dx$ $\newline$ $I = \int (x^4 + 10x^3 + 25x^2) \, dx$
Integrate Each Term: Now we integrate each term separately. $\newline$ $I = \int x^4 \, dx + \int 10x^3 \, dx + \int 25x^2 \, dx$ $\newline$ $I = \frac{1}{5}x^5 + \frac{10}{4}x^4 + \frac{25}{3}x^3 + C$ $\newline$ $I = \frac{1}{5}x^5 + \frac{5}{2}x^4 + \frac{25}{3}x^3 + C$
Simplify Final Answer: We simplify the expression to get the final answer. $I = \frac{1}{5}x^5 + \frac{5}{2}x^4 + \frac{25}{3}x^3 + C$