题目
多项填空题
Question textThe function f(x)=\dfrac{x+2}{(x+1)(x-1)^2} can be resolved into partial fractions of the form:\dfrac{a}{x+1} + \dfrac{b}{x-1} + \dfrac{c}{(x-1)^2}where a,\,b and c are real number values.Use this to express \displaystyle \int f(x)\,dx in the form\displaystyle \int f(x)\,dx = \dfrac{1}{A}\ln|x+1|+\dfrac{1}{B}\ln|x-1| + \dfrac{D}{2x-2}+ C.where A,\,B and D are integer values, and C is a constant of integration.Fill in the correct values for A,\,B, and D.A = Answer 1 Question 23[input] B = Answer 2 Question 23[input] D = Answer 3 Question 23[input]
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思路分析
To tackle this problem, we start by expressing f(x) in partial fractions. The target form uses coefficients a, b, c in the decomposition:
f(x) = a/(x+1) + b/(x-1) + c/(x-1)^2.
From the given integral form, we will relate a, b, c to A, B, D via:
- The integral of a/(x+1) contributes (1/A) ln|x+1|, so a must equal 1/A.
- The integral of b/(x-1) contributes (1/B) ln|x-1|, so b must equal 1/B.
- The integral o......Login to view full explanation登录即可查看完整答案
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Question textThe function f(x)=\dfrac{x+2}{(x+1)(x-1)^2} can be resolved into partial fractions of the form:\dfrac{a}{x+1} + \dfrac{b}{x-1} + \dfrac{c}{(x-1)^2}where a,\,b and c are real number values.Use this to express \displaystyle \int f(x)\,dx in the form\displaystyle \int f(x)\,dx = \dfrac{1}{A}\ln|x+1|+\dfrac{1}{B}\ln|x-1| + \dfrac{D}{2x-2}+ C.where A,\,B and D are integer values, and C is a constant of integration.Fill in the correct values for A,\,B, and D.A = Answer 1 Question 23[input] B = Answer 2 Question 23[input] D = Answer 3 Question 23[input]
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