Question at position 12 Solve ∫x2e2x+1dx\int x^2 e^{2x+1} \, dx.e2x+1(x2−x+12)+Ce^{2x+1}\left( x^2 - x + \frac{1}{2} \right) + Cxe2x+12−e2x+14+C\frac{x e^{2x+1}}{2} - \frac{e^{2x+1}}{4}+ Ce2x+12(x2−x)+C\frac{e^{2x+1}}{2} \left( x^2 - x \right) + Ce2x+12(x2−x+12)+C\frac{e^{2x+1}}{2} \left( x^2 - x + \frac{1}{2} \right) + C

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BlackTom AI · Accepted Answer

To tackle the integral ∫ x^2 e^{2x+1} dx, I first note that e^{2x+1} = e · e^{2x}, so the integral is e · ∫ x^2 e^{2x} dx. The standard approach for ∫ x^n e^{ax} dx yields a polynomial in x multiplied by e^{ax}; here with n = 2 and a = 2, we expect a result of the form e^{2x} times a quadratic in x, up to a constant factor, and then multiplied by the extra e from e^{2x+1}.

Option 1: e^{2x+1} · ( x^2 − x + 1/2 ) + C
- This expression corresponds to a polynomial with coefficients (1, −1, 1/2) inside the parenthesis, which would give an overall coefficient pattern of (1/2)x^2 − (1/2)x + 1/4 after distributing the factor 1 (from inside) and the e^{2x+1} outside. However, the correct polynomial should be (1/2)x^2 − (1/2)x + 1/4 inside the e^{2x+1} factor. The mismatch in the coefficients means this option does not equal the correct antiderivative.

Option 2: x e^{2x+1} + 1/2 − e^{2x+1} + 1/4 + C
- This option mixes terms in a way that does not form a single e^{2x+1} factor multiplying a polynomial in x. The presence of non-multiplicative terms like 1/2 and 1/4 separate from e^{2x+1} indicates it cannot be the antiderivative of x^2 e^{2x+1}. Moreover, the differentiation of such a sum would not produce x^2 e^{2x+1}.

Option 3: e^{2x+1} + 1/2 ( x^2 − x ) + C
- This option places e^{2x+1} outside but only adds a linear combination of a quadratic and a constant in a way that does not align with the expected structure e^{2x+1} times a quadratic. The derivative of this expression would yield terms not matching x^2 e^{2x+1}; specifically, you would get extra e^{2x+1} terms not structured as x^2 e^{2x+1}.

Option 4: e^{2x+1} · 1/2 ( x^2 − x + 1/2 ) + C
- This matches the correct pattern: the integral of x^2 e^{2x+1} dx equals e^{2x+1} times a quadratic with coefficients (1/2) for x^2, (−1/2) for x, and (1/4) for the constant, i.e., (1/2)·(x^2 − x + 1/2). When expanded, this is e^{2x+1}·( (1/2)x^2 − (1/2)x + 1/4 ), which is the standard result obtained via integration by parts or by the general formula ∫ x^n e^{ax} dx.

In summary, the correct antiderivative structure requires e^{2x+1} multiplied by a specific quadratic (with coefficients 1/2, −1/2, and 1/4). Among the options, only Option 4 exactly matches this form, whereas Options 1–3 either misstate the coefficients or fail to present the result in the proper multiplied form.

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