Evaluate the integral. int(4x^(4)+24x^(3)-8x^(2))/(x)dx Answer:

Simplify the integrand: We are given the integral: int((4x^(4)+24x^(3)-8x^(2))/x)dx First, we simplify the integrand by dividing each term by x. (4x^(4)+24x^(3)-8x^(2))/x = 4x^(4-1) + 24x^(3-1) - 8x^(2-1) = 4x^3 + 24x^2 - 8xIntegrate each term: Now we integrate each term separately. int(4x^3)dx + int(24x^2)dx - int(8x)dx Using the power rule for integration, we add 1 to the exponent and divide by the new exponent. int(4x^3)dx = (4/4)x^(3+1) = x^4 int(24x^2)dx = (24/3)x^(2+1) = 8x^3 int(8x)dx = (8/2)x^(1+1) = 4x^2Combine and add constant: Combine the results of the integrations and add the constant of integration C. x^4 + 8x^3 + 4x^2 + C

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Calculus

Find indefinite integrals using the substitution

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Q. Evaluate the integral.

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\int \frac{4 x^{4}+24 x^{3}-8 x^{2}}{x} \mathrm{~d} x

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Answer:

Simplify the integrand: We are given the integral: $\newline$ $\int\frac{4x^{4}+24x^{3}-8x^{2}}{x}\,dx$ $\newline$ First, we simplify the integrand by dividing each term by $x$ . $\newline$ $\frac{4x^{4}+24x^{3}-8x^{2}}{x} = 4x^{4-1} + 24x^{3-1} - 8x^{2-1}$ $\newline$ $= 4x^3 + 24x^2 - 8x$
Integrate each term: Now we integrate each term separately. $\newline$ $\int(4x^3)\,dx + \int(24x^2)\,dx - \int(8x)\,dx$ $\newline$ Using the power rule for integration, we add $1$ to the exponent and divide by the new exponent. $\newline$ $\int(4x^3)\,dx = \left(\frac{4}{4}\right)x^{3+1} = x^4$ $\newline$ $\int(24x^2)\,dx = \left(\frac{24}{3}\right)x^{2+1} = 8x^3$ $\newline$ $\int(8x)\,dx = \left(\frac{8}{2}\right)x^{1+1} = 4x^2$
Combine and add constant: Combine the results of the integrations and add the constant of integration $C$ . $x^4 + 8x^3 + 4x^2 + C$