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iolve the Definite Integral:

int_(-6)^(14)10x^(4)+12x^(3)-16x^(2)-9x+15 dx

iolve the Definite Integral: $\newline$ $\int_{-6}^{14} 10 x^{4}+12 x^{3}-16 x^{2}-9 x+15 d x$

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Evaluate definite integrals using the chain rule

Full solution

Q. iolve the Definite Integral:

\newline

\int_{-6}^{14} 10 x^{4}+12 x^{3}-16 x^{2}-9 x+15 d x

Integrate Terms Separately: Integrate each term separately using the power rule for integrals, which states that the integral of $x^n$ is $(x^{(n+1)})/(n+1)$ for $n \neq -1$ . $\newline$ $\int 10x^4 \, dx = (10/5)x^5 = 2x^5$ $\newline$ $\int 12x^3 \, dx = (12/4)x^4 = 3x^4$ $\newline$ $\int(-16x^2) \, dx = (-16/3)x^3$ $\newline$ $\int(-9x) \, dx = (-9/2)x^2$ $\newline$ $\int 15 \, dx = 15x$
Combine Integrated Terms: Combine the integrated terms to get the antiderivative of the function. $\newline$ Antiderivative: $2x^5 + 3x^4 - \frac{16}{3}x^3 - \frac{9}{2}x^2 + 15x + C$
Evaluate at Upper Limit: Evaluate the antiderivative at the upper limit of integration, $x = 14$ . $F(14) = 2(14)^5 + 3(14)^4 - \left(\frac{16}{3}\right)(14)^3 - \left(\frac{9}{2}\right)(14)^2 + 15(14)$
Evaluate at Lower Limit: Evaluate the antiderivative at the lower limit of integration, $x = -6$ . $F(-6) = 2(-6)^5 + 3(-6)^4 - \left(\frac{16}{3}\right)(-6)^3 - \left(\frac{9}{2}\right)(-6)^2 + 15(-6)$
Subtract Values: Subtract the value of the antiderivative at the lower limit from the value at the upper limit to find the definite integral. $\newline$ $\int_{-6}^{14}$ of the function = $F(14) - F(-6)$
Perform Final Subtraction: Perform the subtraction to get the final answer. $\newline$ Final Answer = $[2(14)^5 + 3(14)^4 - \frac{16}{3}(14)^3 - \frac{9}{2}(14)^2 + 15(14)] - [2(-6)^5 + 3(-6)^4 - \frac{16}{3}(-6)^3 - \frac{9}{2}(-6)^2 + 15(-6)]$

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