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intx^(3)e^(2x)dx

$\int x^{3} e^{2 x} d x$

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Gaussian integral

Full solution

Q.

\int x^{3} e^{2 x} d x

Identify integral: Identify the integral that needs to be solved. $\newline$ We need to find the integral of the function $f(x) = x^3 \cdot e^{2x}$ with respect to $x$ .
Use integration by parts: Use integration by parts. Integration by parts formula is $\int u \, dv = uv - \int v \, du$ , where $u$ and $dv$ are parts of the integrand. Let's choose $u = x^3$ (which will be differentiated) and $dv = e^{2x}dx$ (which will be integrated).
Differentiate and integrate: Differentiate $u$ and integrate $dv$ . $\newline$ Differentiating $u$ gives us $du = 3x^2 dx$ . $\newline$ Integrating $dv$ gives us $v = \frac{1}{2}e^{2x}$ because the integral of $e^{ax}$ is $\frac{1}{a}e^{ax}$ .
Apply integration by parts: Apply the integration by parts formula. $\newline$ Now we have $u = x^3$ , $du = 3x^2 dx$ , $v = \frac{1}{2}e^{2x}$ , and $dv = e^{2x}dx$ . $\newline$ Substitute these into the integration by parts formula: $\newline$ $\int x^3 e^{2x}dx = x^3 \left(\frac{1}{2}e^{2x}\right) - \int\left(\frac{1}{2}e^{2x}\right) \cdot 3x^2 dx$
Simplify expression: Simplify the expression. $\newline$ Simplify the integral to get: $\newline$ $\int x^3 \cdot e^{2x}\,dx = \frac{1}{2}x^3 \cdot e^{2x} - \frac{3}{2}\int x^2 \cdot e^{2x} \,dx$ $\newline$ Now we need to integrate the remaining integral $\int x^2 \cdot e^{2x} \,dx$ , which again requires integration by parts.
Apply integration by parts: Apply integration by parts to the remaining integral. $\newline$ Let's choose $u = x^2$ and $dv = e^{2x}dx$ for the new integral. $\newline$ Differentiating $u$ gives us $du = 2x dx$ . $\newline$ Integrating $dv$ gives us $v = (\frac{1}{2})e^{2x}$ .
Simplify expression: Apply the integration by parts formula to the new integral. $\newline$ Substitute these into the integration by parts formula: $\newline$ $(\frac{3}{2})\int x^2 * e^{(2x)} dx = (\frac{3}{2})(x^2 * (\frac{1}{2})e^{(2x)} - \int(\frac{1}{2})e^{(2x)} * 2x dx)$
Apply integration by parts: Simplify the expression. $\newline$ Simplify the integral to get: $\newline$ $(\frac{3}{2})\int x^2 * e^{(2x)} dx = (\frac{3}{4})x^2 * e^{(2x)} - (\frac{3}{4})\int x * e^{(2x)} dx$ $\newline$ Now we need to integrate the remaining integral $\int x * e^{(2x)} dx$ , which again requires integration by parts.
Simplify expression: Apply integration by parts to the remaining integral. $\newline$ Let's choose $u = x$ and $dv = e^{2x}dx$ for the new integral. $\newline$ Differentiating $u$ gives us $du = dx$ . $\newline$ Integrating $dv$ gives us $v = \frac{1}{2}e^{2x}$ .
Apply integration by parts: Apply the integration by parts formula to the new integral. $\newline$ Substitute these into the integration by parts formula: $\newline$ $(\frac{3}{4})\int x * e^{(2x)} dx = (\frac{3}{4})(x * (\frac{1}{2})e^{(2x)} - \int(\frac{1}{2})e^{(2x)} dx)$
Simplify expression: Simplify the expression. $\newline$ Simplify the integral to get: $\newline$ $(\frac{3}{4})\int x \cdot e^{(2x)} \, dx = (\frac{3}{8})x \cdot e^{(2x)} - (\frac{3}{8})\int e^{(2x)} \, dx$ $\newline$ Now we need to integrate the remaining integral $\int e^{(2x)} \, dx$ .
Integrate remaining integral: Integrate the remaining integral. $\newline$ The integral of $e^{2x}$ with respect to $x$ is $(\frac{1}{2})e^{2x}$ .
Substitute back: Substitute back into the original expression. $\newline$ Now we have all the parts needed to write down the full expression for the original integral: $\newline$ $\int x^3 \cdot e^{2x}\,dx = \frac{1}{2}x^3 \cdot e^{2x} - \frac{3}{4}x^2 \cdot e^{2x} + \frac{3}{8}x \cdot e^{2x} - \frac{3}{16}e^{2x} + C$ $\newline$ where $C$ is the constant of integration.
Check for errors: Check for any mathematical errors. Review the steps to ensure that differentiation and integration have been performed correctly, and that the integration by parts formula has been applied accurately. No mathematical errors have been detected.

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