Evaluate the integral. int2x^(2)e^(-x)dx Answer:

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Evaluate the integral.  int2x^(2)e^(-x)dx Answer:

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Choose u and dv: Let's use integration by parts to solve the integral of 2x^2e^(-x) with respect to x. Integration by parts is given by the formula ∫u dv = uv - ∫v du, where u and dv are parts of the integrand that we choose. We will let u = x^2 and dv = 2e^(-x)dx. Then we need to find du and v. First, we calculate du by differentiating u with respect to x: du = d(x^2)/dx = 2xdx. Next, we find v by integrating dv: v = ∫2e^(-x)dx. To integrate v, we use the fact that the integral of e^(-x) is -e^(-x), so: v = -2e^(-x).Apply integration by parts: Now that we have u, du, v, and dv, we can apply the integration by parts formula: ∫2x^2e^(-x)dx = uv - ∫v du = x^2(-2e^(-x)) - ∫(-2e^(-x))(2x)dx = -2x^2e^(-x) - ∫(-4xe^(-x))dx. We now have a new integral to solve, which is ∫(-4xe^(-x))dx. We will use integration by parts again for this new integral.New integral to solve: For the new integral ∫(-4xe^(-x))dx, we will let u = -4x and dv = e^(-x)dx. Then we need to find du and v. First, we calculate du by differentiating u with respect to x: du = d(-4x)/dx = -4dx. Next, we find v by integrating dv: v = ∫e^(-x)dx. To integrate v, we use the fact that the integral of e^(-x) is -e^(-x), so: v = -e^(-x).Apply integration by parts again: Now we apply the integration by parts formula to the new integral: ∫(-4xe^(-x))dx = uv - ∫v du = (-4x)(-e^(-x)) - ∫(-e^(-x))(-4)dx = 4xe^(-x) - ∫4e^(-x)dx. The remaining integral ∫4e^(-x)dx is straightforward to solve, as it is a constant multiple of the integral of e^(-x), which is -e^(-x).Solve remaining integral: We integrate ∫4e^(-x)dx: ∫4e^(-x)dx = 4∫e^(-x)dx = 4(-e^(-x)) = -4e^(-x). Now we can combine all the parts we have integrated: -2x^2e^(-x) - (4xe^(-x) - (-4e^(-x))).Combine integrated parts: Simplify the expression by distributing the negative sign and combining like terms: -2x^2e^(-x) - 4xe^(-x) + 4e^(-x). This is the antiderivative of the original integral. We also need to add the constant of integration, which we will denote as C.Simplify expression: The final answer for the integral is: ∫2x^2e^(-x)dx = -2x^2e^(-x) - 4xe^(-x) + 4e^(-x) + C.

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

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