题目
题目
多项填空题

(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]

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思路分析
The question has two parts (a) and (b), with blank spaces to fill. The provided answers are the intended completions for each blank. Part (a) (i): Evaluate ∫ x e^x dx. A standard approach is integration by parts or recognizing a derivative pattern. If we set up integration by parts with u = x and dv = e^x dx, then du = dx and v = e^x, giving ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C = (x − 1) e^x + C. The given fill [(x - 1)e^{x}] matches exactly this result, so it is correct. Part (a) (ii)......Login to view full explanation

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Question textThe indefinite integral\displaystyle \int e^{-x}\sin(x)\,dxcan be written in the form\dfrac{\cos(x)}{Ae^x} + \dfrac{\sin(x)}{Be^x} + Cwhere A  and B  are integers, and C  is a constant of integration.Use integration by parts (twice) to find A  and B.Fill in the spaces with the correct responses.A= Answer 1 Question 27[input]B= Answer 2 Question 27[input]

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