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∫x2+x-1x2+x-6 dx - Mathematics and Statistics

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Question

$\int \frac{x^{2} + x - 1}{x^{2} + x - 6} d x$

Sum

Solution

Let I = $\int \frac{x^{2} + x - 1}{x^{2} + x - 6} d x$

= $\int \frac{x^{2} + x - 6 + 5}{x^{2} + x - 6} d x$

= $\int [1 + (\frac{5}{x^{2} + x - 6})] d x$

Let $\frac{5}{x^{2} + x - 6} = \frac{5}{(x + 3) (x - 2)}$

= $\frac{A}{x + 3} + \frac{B}{x - 2}$

∴ 5 = A(x − 2) + B(x + 3) ........(i)

Putting x = 2 in (i), we get

5 = B(5)

∴ B = 1

Putting x = −3 in (i), we get

5 = A(− 5)

∴ A = −1

∴ $\frac{5}{(x + 3) (x - 2)} = \frac{- 1}{x + 3} + \frac{1}{x - 2}$

∴ I = $\int [1 + \frac{- 1}{x + 3} + \frac{1}{x - 2}] d x$

= $\int d x - \int \frac{1}{x + 3} d x + \int \frac{1}{x - 2} d x$

= x − log|x + 3| + log|x − 2| + c

∴ I = $x + \log | \frac{x - 2}{x + 3} | + c$

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Methods of Integration: Integration Using Partial Fractions

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Chapter 2.3: Indefinite Integration - Short Answers II

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SCERT Maharashtra Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

Chapter 2.3 Indefinite Integration
Short Answers II | Q 14

2021-2022 (March) Model set 1 shaalaa.com (with solutions)

Q 24 | 3 marks

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