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Q. Evaluate
- Identify type of integral: Identify the type of integral.We are asked to integrate a product of a polynomial function, , and an exponential function, . This suggests that we should use integration by parts.
- Apply integration by parts: Apply the integration by parts formula.The integration by parts formula is , where and are parts of the integrand that we choose. We need to choose and such that and are easy to work with.
- Choose and : Choose and . Let's choose (which means after differentiation) and (which means after integration).
- Differentiate and integrate: Differentiate and integrate .Differentiate to get .Integrate to get $v = \left(\frac{\(1\)}{\(2\)}\right)e^{\(2\)x}.
- Substitute into formula: Substitute \(u\), \(du\), \(v\), and \(dv\) into the integration by parts formula.\(\newline\)Substitute the values into the formula to get:\(\newline\)\(\int x^2 e^{2x} dx = x^2 \cdot (\frac{1}{2})e^{2x} - \int(\frac{1}{2})e^{2x} \cdot 2x dx\)
- Simplify expression: Simplify the expression.\(\newline\)Simplify the integral to get:\(\newline\)\(\int x^2 e^{2x} \, dx = \frac{1}{2}x^2 e^{2x} - \int x e^{2x} \, dx\)
- Apply integration by parts again: Apply integration by parts again to the remaining integral.\(\newline\)We need to apply integration by parts to \(\int x e^{2x} \, dx\). Let's choose \(u = x\) (\(du = dx\)) and \(dv = e^{2x} dx\) (\(v = (1/2)e^{2x}\)).
- Differentiate and integrate: Differentiate \(u\) and integrate \(dv\) for the second application of integration by parts.\(\newline\)Differentiate \(u\) to get \(du = dx\).\(\newline\)Integrate \(dv\) to get \(v = \frac{1}{2}e^{2x}\).
- Substitute into formula again: Substitute \(u\), \(du\), \(v\), and \(dv\) into the integration by parts formula for the second time.\(\newline\)Substitute the values into the formula to get:\(\newline\)\(\int x e^{2x} dx = x \cdot (\frac{1}{2})e^{2x} - \int(\frac{1}{2})e^{2x} dx\)
- Simplify and integrate: Simplify the expression and integrate the remaining term.\(\newline\)Simplify the integral to get:\(\newline\)\(\int x e^{2x} dx = \frac{1}{2}x e^{2x} - \frac{1}{4}e^{2x}\)
- Substitute result back: Substitute the result from Step \(10\) back into the expression from Step \(6\).\(\newline\)Substitute the result to get:\(\newline\)\(\int x^2 e^{2x} dx = \frac{1}{2}x^2 e^{2x} - \left[\frac{1}{2}x e^{2x} - \frac{1}{4}e^{2x}\right]\)
- Simplify final expression: Simplify the final expression.\(\newline\)Simplify the expression to get:\(\newline\)\(\int x^2 e^{2x} dx = \frac{1}{2}x^2 e^{2x} - \frac{1}{2}x e^{2x} + \frac{1}{4}e^{2x} + C\), where \(C\) is the constant of integration.
