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

Question

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.

查看解析

登录即可查看完整答案

类似问题

(a) Integrate the following (i) (Hint: You may substitute , and adopt integration by parts) [2 marks](ii) (Hint: You may let , , use and adopt integration by parts)[2 marks] (b) Differentiate [3 marks][Fill in the blank]

Compute [math: ∫01(2x2+3x−2)ex dx]\displaystyle \int _0^1{(2x^2+3x-2)e^x dx}.

Question at position 9 ∫033xex3dx=\int_0^33xe^{\frac{x}{3}}dx=903276

Use of integration by parts \displaystyle\int f(x)\,g'(x)\,dx=f(x)\,g(x) - \displaystyle \int f'(x)\,g(x)\,dxrequires a suitable choice for f(x) and g'(x).Choose from the options below, the choice of f(x) and g'(x) that will NOT help find the corresponding indefinite integral.

Integration by parts [math: ∫f′(x)g(x)dx=f(x)g(x)−∫f(x)g′(x)dx]\int f'(x) g(x) \, dx = f(x) g(x) - \int f(x) g'(x) \, dx is the partial integral counterpart to which of the following rules?

Integration by parts \[\int f'(x) g(x) \, dx = f(x) g(x) - \int f(x) g'(x) \, dx \] is the partial integral counterpart to which of the following rules?

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单