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F(x)=int_(-2)^(2x)(3t^(2)+2t)dt

F^(')(x)=

$F(x)=\int_{-2}^{2 x}\left(3 t^{2}+2 t\right) d t$ $\newline$ $F^{\prime}(x)=$

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Evaluate definite integrals using the power rule

Full solution

Q.

F(x)=\int_{-2}^{2 x}\left(3 t^{2}+2 t\right) d t

\newline

F^{\prime}(x)=

Use Fundamental Theorem: Use the Fundamental Theorem of Calculus Part $1$ , which states that if $F(x)$ is defined as the integral from $a$ to $x$ of $f(t)$ dt, then $F'(x)$ is $f(x)$ .
Differentiate Upper Limit: Differentiate the upper limit of the integral, which is $2x$ , with respect to $x$ . The derivative of $2x$ is $2$ .
Plug Upper Limit: Plug the upper limit into the integrand and multiply by the derivative of the upper limit. So, $F'(x) = 2 \times (3(2x)^2 + 2(2x))$ .
Simplify Expression: Simplify the expression. $F'(x) = 2 \times (12x^2 + 4x) = 24x^2 + 8x$ .

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