Evaluate the integral. int xe^(2x)dx Answer:

Choose u and dv: To solve the integral of xe^(2x) with respect to x, we will use integration by parts, which is given by the formula ∫u dv = uv - ∫v du, where u and dv are parts of the integrand that we choose.Apply integration by parts: Let's choose u = x (which will be differentiated) and dv = e^(2x)dx (which will be integrated). Differentiating u gives us du = dx, and integrating dv gives us v = (1/2)e^(2x).Integrate dv: Now we apply the integration by parts formula: ∫xe^(2x)dx = uv - ∫v du. Substituting the chosen parts, we get ∫xe^(2x)dx = x*(1/2)e^(2x) - ∫(1/2)e^(2x)dx.Substitute back: Next, we integrate (1/2)e^(2x) with respect to x, which gives us (1/2)*(1/2)e^(2x) = (1/4)e^(2x).Final result: Substituting this back into our equation, we have ∫xe^(2x)dx = (1/2)xe^(2x) - (1/4)e^(2x) + C, where C is the constant of integration.Therefore, the integral of xe^(2x) with respect to x is (1/2)xe^(2x) - (1/4)e^(2x) + C.

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Evaluate the integral.

int xe^(2x)dx
Answer:

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

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Calculus

Find indefinite integrals using the substitution and by parts

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Q. Evaluate the integral.

\newline

\int x e^{2 x} d x

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Answer:

Choose $u$ and $dv$ : To solve the integral of $xe^{2x}$ with respect to $x$ , we will use integration by parts, which is given by the formula $\int u\, dv = uv - \int v\, du$ , where $u$ and $dv$ are parts of the integrand that we choose.
Apply integration by parts: Let's choose $u = x$ (which will be differentiated) and $dv = e^{2x}dx$ (which will be integrated). Differentiating $u$ gives us $du = dx$ , and integrating $dv$ gives us $v = \frac{1}{2}e^{2x}$ .
Integrate dv: Now we apply the integration by parts formula: $\int x e^{2x} dx = uv - \int v du$ . Substituting the chosen parts, we get $\int x e^{$2$x} dx = x \left(\frac{$1$}{$2$}\right)e^{$2$x} - \int \left(\frac{$1$}{$2$}\right)e^{$2$x} dx.
Substitute back: Next, we integrate $(1/2)e^{2x}$ with respect to $x$, which gives us $(1/2)\cdot(1/2)e^{2x} = (1/4)e^{2x}$.
Final result: Substituting this back into our equation, we have $\int x e^{2x} dx = \left(\frac{1}{2}\right) x e^{2x} - \left(\frac{1}{4}\right) e^{2x} + C$, where $C$ is the constant of integration.
Final result: Substituting this back into our equation, we have $\int xe^{2x}\,dx = \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} + C$, where $C$ is the constant of integration.Therefore, the integral of $xe^{2x}$ with respect to $x$ is $\frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} + C$.