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

Simplify integrand: We start by simplifying the integrand by dividing each term by x^3. (2x^4 - 3x^3 + x) / x^3 = 2x^(4-3) - 3x^(3-3) + x^(1-3) = 2x - 3 + x^(-2)Integrate simplified terms: Now we integrate each term separately. ∫(2x - 3 + x^(-2))dx = ∫2xdx - ∫3dx + ∫x^(-2)dxCombine integrated terms: Integrate each term. ∫2xdx = 2 * (1/2) * x^2 = x^2 ∫3dx = 3x ∫x^(-2)dx = x^(-2+1)/(-2+1) = -x^(-1)Add constant of integration: Combine the integrated terms and add the constant of integration C. x^2 - 3x - x^(-1) + CFinal answer notation: Rewrite the final answer in a more conventional notation, replacing x^(-1) with 1/x. x^2 - 3x - 1/x + 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{2 x^{4}-3 x^{3}+x}{x^{3}} \mathrm{~d} x

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Answer:

Simplify integrand: We start by simplifying the integrand by dividing each term by $x^3$ . $\newline$ $(2x^4 - 3x^3 + x) / x^3 = 2x^{4-3} - 3x^{3-3} + x^{1-3}$ $\newline$ $= 2x - 3 + x^{-2}$
Integrate simplified terms: Now we integrate each term separately. $\newline$ \int(\(2x - $3$ + x^{ $-2$ })\,dx = \int $2$ x\,dx - \int $3$ \,dx + \int x^{ $-2$ }\,dx
Combine integrated terms: Integrate each term. $\newline$ $\int 2x\,dx = 2 \times \frac{1}{2} \times x^2 = x^2$ $\newline$ $\int 3\,dx = 3x$ $\newline$ $\int x^{-2}\,dx = \frac{x^{-2+1}}{-2+1} = -x^{-1}$
Add constant of integration: Combine the integrated terms and add the constant of integration $C$ . $x^2 - 3x - x^{-1} + C$
Final answer notation: Rewrite the final answer in a more conventional notation, replacing $x^{-1}$ with $\frac{1}{x}$ . $x^2 - 3x - \frac{1}{x} + C$