Evaluate the integral. int6x^(3)e^(-2x)dx Answer:

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Evaluate the integral.  int6x^(3)e^(-2x)dx Answer:

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Identify integral: Identify the integral to be solved. We need to evaluate the integral of the function 6x^(3)e^(-2x) with respect to x. This is an integration problem that requires the use of integration by parts.Apply integration by parts: Apply integration by parts. Integration by parts is given by the formula ∫u dv = uv - ∫v du, where u and dv are parts of the integrand. We choose u = x^3 (which will be differentiated) and dv = 6e^(-2x)dx (which will be integrated).Differentiate and integrate: Differentiate u and integrate dv. Differentiating u gives us du = 3x^2 dx. Integrating dv gives us v = -3e^(-2x) (since the integral of e^(-2x) is -1/2 e^(-2x) and we have a factor of 6).Substitute into formula: Substitute u, v, du, and dv into the integration by parts formula. Now we have all the components to apply integration by parts: ∫6x^(3)e^(-2x)dx = uv - ∫v du = (-3e^(-2x))(x^3) - ∫(-3e^(-2x))(3x^2)dx = -3x^3e^(-2x) + 9∫x^2e^(-2x)dxApply integration by parts again: Apply integration by parts again to the remaining integral. We need to integrate 9∫x^2e^(-2x)dx. We choose u = x^2 and dv = 9e^(-2x)dx. Differentiating u gives us du = 2x dx. Integrating dv gives us v = -9/2 e^(-2x).Substitute into formula again: Substitute u, v, du, and dv into the integration by parts formula for the second time. ∫9x^2e^(-2x)dx = uv - ∫v du = (-9/2 e^(-2x))(x^2) - ∫(-9/2 e^(-2x))(2x)dx = -9/2 x^2e^(-2x) + 9∫xe^(-2x)dxIntegrate exponential function: Apply integration by parts for the third time to the remaining integral. We need to integrate 9∫xe^(-2x)dx. We choose u = x and dv = 9e^(-2x)dx. Differentiating u gives us du = dx. Integrating dv gives us v = -9/2 e^(-2x).Combine all parts: Substitute u, v, du, and dv into the integration by parts formula for the third time. ∫9xe^(-2x)dx = uv - ∫v du = (-9/2 e^(-2x))(x) - ∫(-9/2 e^(-2x))dx = -9/2 xe^(-2x) + 9/2 ∫e^(-2x)dxIntegrate the remaining exponential function. The integral of e^(-2x) is -1/2 e^(-2x), so we have: 9/2 ∫e^(-2x)dx = -9/4 e^(-2x)Combine all parts to get the final answer. Now we combine all the parts from the previous steps: ∫6x^(3)e^(-2x)dx = -3x^3e^(-2x) - 9/2 x^2e^(-2x) - 9/2 xe^(-2x) - 9/4 e^(-2x) + C This is the final answer, where C is the constant of integration.

Evaluate the integral. $\newline$ $\int 6 x^{3} e^{-2 x} d x$ $\newline$ Answer:

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Evaluate the integral. $\newline$ $\int 6 x^{3} e^{-2 x} d x$ $\newline$ Answer: