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Evaluate the following integral using integration by parts. \[ \int x^{2} e^{12 x} d x \]

Ask by Johnston Mcguire. in the United States
Mar 17,2025

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The integral of \( x^{2} e^{12x} \) with respect to \( x \) is: \[ \frac{e^{12x}x^{2}}{12} - \frac{e^{12x}x}{72} + \frac{e^{12x}}{864} + C \] where \( C \) is the constant of integration.

Solución

Calculate the integral \( \int x^{2} e^{12x} dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x^{2}e^{12x} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x^{2}\\&dv=e^{12x}dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=2xdx\\&dv=e^{12x}dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=2xdx\\&v=\frac{e^{12x}}{12}\end{align}\) - step4: Substitute the values into formula: \(x^{2}\times \frac{e^{12x}}{12}-\int 2x\times \frac{e^{12x}}{12} dx\) - step5: Calculate: \(\frac{x^{2}e^{12x}}{12}-\int \frac{xe^{12x}}{6} dx\) - step6: Evaluate the integral: \(\frac{x^{2}e^{12x}}{12}-\frac{1}{6}\times \frac{xe^{12x}}{12}+\frac{e^{12x}}{864}\) - step7: Calculate: \(\frac{x^{2}e^{12x}}{12}-\frac{xe^{12x}}{72}+\frac{e^{12x}}{864}\) - step8: Simplify the expression: \(\frac{e^{12x}x^{2}}{12}-\frac{xe^{12x}}{72}+\frac{e^{12x}}{864}\) - step9: Simplify the expression: \(\frac{e^{12x}x^{2}}{12}-\frac{e^{12x}x}{72}+\frac{e^{12x}}{864}\) - step10: Add the constant of integral C: \(\frac{e^{12x}x^{2}}{12}-\frac{e^{12x}x}{72}+\frac{e^{12x}}{864} + C, C \in \mathbb{R}\) The integral of \(x^{2} e^{12x}\) with respect to \(x\) is given by: \[ \int x^{2} e^{12x} dx = \frac{e^{12x}x^{2}}{12} - \frac{e^{12x}x}{72} + \frac{e^{12x}}{864} + C \] where \(C\) is the constant of integration.

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To evaluate the integral \(\int x^{2} e^{12 x} d x\) using integration by parts, we apply the integration by parts formula, which is given by: \[ \int u \, dv = uv - \int v \, du \] We start by choosing: - \(u = x^2\) which means \(du = 2x \, dx\) - \(dv = e^{12x} \, dx\) which gives us \(v = \frac{1}{12} e^{12x}\) Now we can apply the integration by parts formula: \[ \int x^{2} e^{12x} \, dx = uv - \int v \, du \] Substituting the values we found: \[ = x^2 \cdot \frac{1}{12} e^{12x} - \int \frac{1}{12} e^{12x} \cdot 2x \, dx \] This simplifies to: \[ = \frac{x^2}{12} e^{12x} - \frac{1}{6} \int x e^{12x} \, dx \] Now, we need to solve \(\int x e^{12x} \, dx\) using integration by parts again. Let’s choose: - \(u = x\) which means \(du = dx\) - \(dv = e^{12x} \, dx\) which means \(v = \frac{1}{12} e^{12x}\) Applying the integration by parts formula again: \[ \int x e^{12x} \, dx = uv - \int v \, du \] Substituting the values: \[ = x \cdot \frac{1}{12} e^{12x} - \int \frac{1}{12} e^{12x} \cdot 1 \, dx \] This simplifies to: \[ = \frac{x}{12} e^{12x} - \frac{1}{12} \int e^{12x} \, dx \] The integral of \(e^{12x}\) is: \[ \int e^{12x} \, dx = \frac{1}{12} e^{12x} + C \] So we have: \[ \int x e^{12x} \, dx = \frac{x}{12} e^{12x} - \frac{1}{144} e^{12x} + C \] Now substituting back into our previous expression for \(\int x^2 e^{12x} \, dx\): \[ \int x^2 e^{12x} \, dx = \frac{x^2}{12} e^{12x} - \frac{1}{6} \left( \frac{x}{12} e^{12x} - \frac{1}{144} e^{12x} \right) \] This simplifies to: \[ = \frac{x^2}{12} e^{12x} - \frac{1}{72} x e^{12x} + \frac{1}{864} e^{12x} + C \] Combining everything, we have the final result: \[ \int x^{2} e^{12x} \, dx = \frac{e^{12x}}{864} (72x^2 - 12x + 1) + C \]

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