Evaluate the integral. int-4x^(2)e^(4x)dx Answer:

Identify integral: Identify the integral that needs to be solved. We need to find the indefinite integral of the function -4x^2e^(4x) with respect to x.Use integration by parts: Use integration by parts. Integration by parts formula is ∫u dv = uv - ∫v du. Let u = x^2 (which implies du = 2x dx) and dv = -4e^(4x) dx (which implies v = -e^(4x)).Differentiate and integrate: Differentiate u and integrate dv. Differentiating u gives us du = 2x dx. Integrating dv gives us v = -e^(4x)/4.Apply integration by parts: Apply the integration by parts formula. ∫-4x^2e^(4x) dx = uv - ∫v du = x^2(-e^(4x)/4) - ∫(-e^(4x)/4)(2x dx) = -x^2e^(4x)/4 + (1/2)∫xe^(4x) dxApply integration by parts again: Apply integration by parts again to the remaining integral. For the integral ∫xe^(4x) dx, let u = x (which implies du = dx) and dv = e^(4x) dx (which implies v = e^(4x)/4).Differentiate and integrate: Differentiate u and integrate dv for the second application of integration by parts. Differentiating u gives us du = dx. Integrating dv gives us v = e^(4x)/4.Apply integration by parts: Apply the integration by parts formula to the remaining integral. ∫xe^(4x) dx = uv - ∫v du = x(e^(4x)/4) - ∫(e^(4x)/4) dx = x(e^(4x)/4) - (e^(4x)/16) + CSubstitute and simplify: Substitute the result from Step 7 back into the equation from Step 4. - x^2e^(4x)/4 + (1/2)(x(e^(4x)/4) - (e^(4x)/16) + C) = - x^2e^(4x)/4 + x(e^(4x)/8) - (e^(4x)/32) + C/2Combine the constants and simplify the expression. The final answer is the indefinite integral of the given function: ∫-4x^2e^(4x) dx = - x^2e^(4x)/4 + x(e^(4x)/8) - (e^(4x)/32) + C

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Evaluate the integral.

int-4x^(2)e^(4x)dx
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Evaluate the integral. $\newline$ $\int-4 x^{2} e^{4 x} d x$ $\newline$ Answer:

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate the integral.

\newline

\int-4 x^{2} e^{4 x} d x

\newline

Answer:

Identify integral: Identify the integral that needs to be solved. $\newline$ We need to find the indefinite integral of the function $-4x^2e^{4x}$ with respect to $x$ .
Use integration by parts: Use integration by parts. $\newline$ Integration by parts formula is $\int u \, dv = uv - \int v \, du$ . $\newline$ Let $u = x^2$ (which implies $du = 2x \, dx$ ) and $dv = -4e^{4x} \, dx$ (which implies $v = -e^{4x}$ ).
Differentiate and integrate: Differentiate $u$ and integrate $dv$ . Differentiating $u$ gives us $du = 2x \, dx$ . Integrating $dv$ gives us $v = -\frac{e^{4x}}{4}.$
Apply integration by parts: Apply the integration by parts formula. $\newline$ $\int -4x^2e^{4x} dx = uv - \int v du$ $\newline$ = $x^2(-e^{4x}/4) - \int(-e^{4x}/4)(2x dx)$ $\newline$ = $-x^2e^{4x}/4 + (1/2)\int xe^{4x} dx$
Apply integration by parts again: Apply integration by parts again to the remaining integral. $\newline$ For the integral $\int x e^{4x} \, dx$ , let $u = x$ (which implies $du = dx$ ) and $dv = e^{4x} dx$ (which implies $v = \frac{e^{4x}}{4}$ ).
Differentiate and integrate: Differentiate $u$ and integrate $dv$ for the second application of integration by parts. $\newline$ Differentiating $u$ gives us $du = dx$ . $\newline$ Integrating $dv$ gives us $v = \frac{e^{4x}}{4}$ .
Apply integration by parts: Apply the integration by parts formula to the remaining integral. $\newline$ $\int x e^{4x} \, dx = uv - \int v \, du$ $\newline$ = $x\left(\frac{e^{4x}}{4}\right) - \int\left(\frac{e^{4x}}{4}\right) \, dx$ $\newline$ = $x\left(\frac{e^{4x}}{4}\right) - \left(\frac{e^{4x}}{16}\right) + C$
Substitute and simplify: Substitute the result from Step $7$ back into the equation from Step $4$ . $\newline$ $- \frac{x^2e^{4x}}{4} + \frac{1}{2}\left(x\left(\frac{e^{4x}}{4}\right) - \frac{e^{4x}}{16} + C\right) = - \frac{x^2e^{4x}}{4} + \frac{x(e^{4x})}{8} - \frac{e^{4x}}{32} + \frac{C}{2}$
Substitute and simplify: Substitute the result from Step $7$ back into the equation from Step $4$ . $\newline$ $- \frac{x^2e^{4x}}{4} + \frac{1}{2}\left(\frac{x(e^{4x})}{4} - \frac{e^{4x}}{16} + C\right)$ $\newline$ = $- \frac{x^2e^{4x}}{4} + \frac{x(e^{4x})}{8} - \frac{e^{4x}}{32} + \frac{C}{2}$ Combine the constants and simplify the expression. $\newline$ The final answer is the indefinite integral of the given function: $\newline$ $\int -4x^2e^{4x} dx = - \frac{x^2e^{4x}}{4} + \frac{x(e^{4x})}{8} - \frac{e^{4x}}{32} + C$