Evaluate the integral. int(15x^(3)+2x^(2)-8x)/(x)dx Answer:

Simplify integrand: Simplify the integrand by dividing each term by x. (15x^(3)+2x^(2)-8x)/(x) = 15x^(3)/x + 2x^(2)/x - 8x/x = 15x^(2) + 2x - 8Integrate each term: Integrate each term separately. ∫(15x^(2) + 2x - 8)dx = ∫15x^(2)dx + ∫2xdx - ∫8dx = 15∫x^(2)dx + 2∫xdx - 8∫dxCalculate integrals: Calculate the integral of each term. 15∫x^(2)dx = 15 * (x^(2+1)/(2+1)) = 15 * (x^3/3) 2∫xdx = 2 * (x^(1+1)/(1+1)) = 2 * (x^2/2) 8∫dx = 8xCombine results: Combine the results and add the constant of integration C. 15 * (x^3/3) + 2 * (x^2/2) + 8x + C = 5x^3 + x^2 + 8x + 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{15 x^{3}+2 x^{2}-8 x}{x} \mathrm{~d} x

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Answer:

Simplify integrand: Simplify the integrand by dividing each term by $x$ . $\frac{15x^{3}+2x^{2}-8x}{x} = \frac{15x^{3}}{x} + \frac{2x^{2}}{x} - \frac{8x}{x}$ $= 15x^{2} + 2x - 8$
Integrate each term: Integrate each term separately. $\newline$ $\int(15x^{2} + 2x - 8)\,dx = \int 15x^{2}\,dx + \int 2x\,dx - \int 8\,dx$ $\newline$ = $15\int x^{2}\,dx + 2\int x\,dx - 8\int dx$
Calculate integrals: Calculate the integral of each term. $\newline$ $15\int x^{2}\,dx = 15 \times \left(\frac{x^{2+1}}{2+1}\right) = 15 \times \left(\frac{x^3}{3}\right)$ $\newline$ $2\int x\,dx = 2 \times \left(\frac{x^{1+1}}{1+1}\right) = 2 \times \left(\frac{x^2}{2}\right)$ $\newline$ $8\int \,dx = 8x$
Combine results: Combine the results and add the constant of integration $C$ . $15 \times \left(\frac{x^3}{3}\right) + 2 \times \left(\frac{x^2}{2}\right) + 8x + C$ $= 5x^3 + x^2 + 8x + C$