Solve the Definite integral int_(2)^(3)[8x^(3)+3x^(2)+6x]

Set up integral: Set up the integral for the given function. We need to evaluate the definite integral of the function 8x^3 + 3x^2 + 6x from x = 2 to x = 3. The integral is written as: ∫(from x=2 to x=3) (8x^3 + 3x^2 + 6x) dxApply power rule: Apply the power rule for integration to each term of the function. The power rule states that the integral of x^n is (x^(n+1))/(n+1) for any real number n that is not equal to -1. Using this rule, we find the antiderivative of each term: ∫8x^3 dx = 8 * (x^(3+1))/(3+1) = 8 * (x^4)/4 = 2x^4 ∫3x^2 dx = 3 * (x^(2+1))/(2+1) = 3 * (x^3)/3 = x^3 ∫6x dx = 6 * (x^(1+1))/(1+1) = 6 * (x^2)/2 = 3x^2Combine antiderivatives: Combine the antiderivatives to get the indefinite integral. The indefinite integral of the function is the sum of the antiderivatives of its terms: ∫(8x^3 + 3x^2 + 6x) dx = 2x^4 + x^3 + 3x^2 + C where C is the constant of integration.Evaluate indefinite integral: Evaluate the indefinite integral from x = 2 to x = 3. To find the definite integral, we substitute the upper and lower limits of integration into the indefinite integral and subtract: (2(3)^4 + (3)^3 + 3(3)^2) - (2(2)^4 + (2)^3 + 3(2)^2) = (2(81) + 27 + 3(9)) - (2(16) + 8 + 3(4)) = (162 + 27 + 27) - (32 + 8 + 12) = 216 - 52Perform final calculation: Perform the final calculation to obtain the value of the definite integral. 216 - 52 = 164

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Solve the Definite integral

int_(2)^(3)[8x^(3)+3x^(2)+6x]

Solve the Definite integral $\newline$ $\int_{2}^{3}\left[8 x^{3}+3 x^{2}+6 x\right]$

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Evaluate definite integrals using the chain rule

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Q. Solve the Definite integral

\newline

\int_{2}^{3}\left[8 x^{3}+3 x^{2}+6 x\right]

Set up integral: Set up the integral for the given function. $\newline$ We need to evaluate the definite integral of the function $8x^3 + 3x^2 + 6x$ from $x = 2$ to $x = 3$ . $\newline$ The integral is written as: $\newline$ $\int_{x=2}^{x=3} (8x^3 + 3x^2 + 6x) \, dx$
Apply power rule: Apply the power rule for integration to each term of the function. $\newline$ The power rule states that the integral of $x^n$ is $(x^{(n+1)})/(n+1)$ for any real number $n$ that is not equal to $-1$ . $\newline$ Using this rule, we find the antiderivative of each term: $\newline$ $\int 8x^3 \, dx = 8 \cdot (x^{(3+1)})/(3+1) = 8 \cdot (x^4)/4 = 2x^4$ $\newline$ $\int 3x^2 \, dx = 3 \cdot (x^{(2+1)})/(2+1) = 3 \cdot (x^3)/3 = x^3$ $\newline$ $\int 6x \, dx = 6 \cdot (x^{(1+1)})/(1+1) = 6 \cdot (x^2)/2 = 3x^2$
Combine antiderivatives: Combine the antiderivatives to get the indefinite integral. $\newline$ The indefinite integral of the function is the sum of the antiderivatives of its terms: $\newline$ $\int(8x^3 + 3x^2 + 6x) \, dx = 2x^4 + x^3 + 3x^2 + C$ $\newline$ where $C$ is the constant of integration.
Evaluate indefinite integral: Evaluate the indefinite integral from $x = 2$ to $x = 3$ . To find the definite integral, we substitute the upper and lower limits of integration into the indefinite integral and subtract: $(2(3)^4 + (3)^3 + 3(3)^2) - (2(2)^4 + (2)^3 + 3(2)^2) = (2(81) + 27 + 3(9)) - (2(16) + 8 + 3(4)) = (162 + 27 + 27) - (32 + 8 + 12) = 216 - 52$
Perform final calculation: Perform the final calculation to obtain the value of the definite integral. $216 - 52 = 164$