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Evaluate the integral.

int2x^(2)e^(-x)dx
Answer:

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

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Find indefinite integrals using the substitution and by parts

Full solution

Q. Evaluate the integral.

\newline

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

\newline

Answer:

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 $\int u dv = uv - \int v du$ , 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 $dv$ $0$ . $\newline$ First, we calculate $du$ by differentiating $u$ with respect to $x$ : $\newline$ $dv$ $4$ . $\newline$ Next, we find $dv$ $0$ by integrating $dv$ : $\newline$ $dv$ $7$ . $\newline$ To integrate $dv$ $0$ , we use the fact that the integral of $dv$ $9$ is $2x^2e^{-x}$ $0$ , so: $\newline$ $2x^2e^{-x}$ $1$ .
Apply integration by parts: Now that we have $u$ , $du$ , $v$ , and $dv$ , we can apply the integration by parts formula: $\newline$ $\int 2x^2e^{(-x)}dx = uv - \int v du$ $\newline$ $= x^2(-2e^{(-x)}) - \int(-2e^{(-x)})(2x)dx$ $\newline$ $= -2x^2e^{(-x)} - \int(-4xe^{(-x)})dx.$ $\newline$ We now have a new integral to solve, which is $\int(-4xe^{(-x)})dx$ . We will use integration by parts again for this new integral.
New integral to solve: For the new integral $\int(-4xe^{-x})dx$ , we will let $u = -4x$ and $dv = e^{-x}dx$ . Then we need to find $du$ and $v$ . $\newline$ First, we calculate $du$ by differentiating $u$ with respect to $x$ : $\newline$ $du = d(-4x)/dx = -4dx$ . $\newline$ Next, we find $v$ by integrating $u = -4x$ $0$ : $\newline$ $u = -4x$ $1$ . $\newline$ To integrate $v$ , we use the fact that the integral of $u = -4x$ $3$ is $u = -4x$ $4$ , so: $\newline$ $u = -4x$ $5$ .
Apply integration by parts again: Now we apply the integration by parts formula to the new integral: $\newline$ $\int(-4xe^{-x})dx = uv - \int v du$ $\newline$ = $(-4x)(-e^{-x}) - \int(-e^{-x})(-4)dx$ $\newline$ = $4xe^{-x} - \int 4e^{-x}dx$ . $\newline$ The remaining integral $\int 4e^{-x}dx$ 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 $\int 4e^{-x}\,dx$ : $\newline$ $\int 4e^{-x}\,dx = 4\int e^{-x}\,dx$ $\newline$ $= 4(-e^{-x})$ $\newline$ $= -4e^{-x}.$ $\newline$ Now we can combine all the parts we have integrated: $\newline$ $-2x^2e^{-x} - (4xe^{-x} - (-4e^{-x})).$
Combine integrated parts: Simplify the expression by distributing the negative sign and combining like terms: $\newline$ $-2x^2e^{-x} - 4xe^{-x} + 4e^{-x}$ . $\newline$ 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: $\int 2x^2e^{-x}\,dx = -2x^2e^{-x} - 4xe^{-x} + 4e^{-x} + C$ .

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