Evaluate int_(-3)^(4)(2kx+3kx^(2))dx giving your answer in terms of k.

Write Integral Function: Write down the integral that needs to be evaluated. We need to evaluate the definite integral of the function (2kx + 3kx^2) from x = -3 to x = 4. ∫(-3 to 4) (2kx + 3kx^2) dxApply Linearity: Apply the linearity of the integral to split the integral into two separate integrals. ∫(-3 to 4) (2kx + 3kx^2) dx = ∫(-3 to 4) 2kx dx + ∫(-3 to 4) 3kx^2 dxEvaluate First Integral: Evaluate the first integral ∫(-3 to 4) 2kx dx. The antiderivative of 2kx with respect to x is kx^2. We will evaluate this antiderivative from -3 to 4. kx^2 | from x = -3 to x = 4 = k(4^2) - k(-3^2) = 16k - 9k = 7kEvaluate Second Integral: Evaluate the second integral ∫(-3 to 4) 3kx^2 dx. The antiderivative of 3kx^2 with respect to x is kx^3. We will evaluate this antiderivative from -3 to 4. kx^3 | from x = -3 to x = 4 = k(4^3) - k(-3^3) = 64k - (-27k) = 64k + 27k = 91kCombine Results: Combine the results from Step 3 and Step 4 to get the final answer. The final answer is the sum of the two evaluated integrals: 7k + 91k = 98k

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Evaluate

int_(-3)^(4)(2kx+3kx^(2))dx
giving your answer in terms of
k.

Evaluate $\newline$ $\int_{-3}^{4}\left(2 k x+3 k x^{2}\right) d x$ $\newline$ giving your answer in terms of $k$ .

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate

\newline

\int_{-3}^{4}\left(2 k x+3 k x^{2}\right) d x

\newline

giving your answer in terms of

k

Write Integral Function: Write down the integral that needs to be evaluated. $\newline$ We need to evaluate the definite integral of the function $2kx + 3kx^2$ from $x = -3$ to $x = 4$ . $\newline$ $\int_{-3}^{4} (2kx + 3kx^2) \, dx$
Apply Linearity: Apply the linearity of the integral to split the integral into two separate integrals. $\newline$ $\int_{-3}^{4} (2kx + 3kx^2) \, dx = \int_{-3}^{4} 2kx \, dx + \int_{-3}^{4} 3kx^2 \, dx$
Evaluate First Integral: Evaluate the first integral $\int_{-3}^{4} 2kx \, dx$ . The antiderivative of $2kx$ with respect to $x$ is $kx^2$ . We will evaluate this antiderivative from $-3$ to $4$ . $kx^2 |_{x = -3}^{x = 4} = k(4^2) - k(-3^2) = 16k - 9k = 7k$
Evaluate Second Integral: Evaluate the second integral $\int_{-3}^{4} 3kx^2 dx$ . The antiderivative of $3kx^2$ with respect to $x$ is $kx^3$ . We will evaluate this antiderivative from $-3$ to $4$ . $kx^3 |_{x = -3}^{x = 4} = k(4^3) - k(-3^3) = 64k - (-27k) = 64k + 27k = 91k$
Combine Results: Combine the results from Step $3$ and Step $4$ to get the final answer. $\newline$ The final answer is the sum of the two evaluated integrals: $\newline$ $7k + 91k = 98k$