Theory of Intelligence
11TH•Mathematics•Partial Fractions•Exercise 5.1

Introduction

Practice set 5.1 (partial fractions: distinct/repeated linear factors, irreducible quadratics, and improper fractions).

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Q1

Question 1

Resolve into partial fractions: 2x2−1\dfrac{2}{x^2-1}

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Step-by-Step Solution

Question 1

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Solution

2x2−1=2(x−1)(x+1)\frac{2}{x^2-1}=\frac{2}{(x-1)(x+1)}

Let:

2(x−1)(x+1)=Ax−1+Bx+1\frac{2}{(x-1)(x+1)}=\frac{A}{x-1}+\frac{B}{x+1}

Multiply by (x−1)(x+1)(x-1)(x+1):

2=A(x+1)+B(x−1)2=A(x+1)+B(x-1)

Put x=1x=1:

2=2A⇒A=12=2A\Rightarrow A=1

Put x=−1x=-1:

2=−2B⇒B=−12=-2B\Rightarrow B=-1

Hence:

2x2−1=1x−1−1x+1\boxed{\frac{2}{x^2-1}=\frac{1}{x-1}-\frac{1}{x+1}}
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