First slide

Methods of integration

Question

$Evaluate \int \frac{x^{2} + 1}{(x - 1)^{2} (x + 3)} d x$

Moderate

A

$\frac{3}{8} \log | x - 1 | + \frac{1}{2 (x - 1)} + \frac{5}{8} \log | x + 3 | + C$
B

$\frac{3}{8} \log | x - 1 | - \frac{1}{2 (x - 1)} + \frac{5}{8} \log | x + 3 | + C$
C

$\frac{3}{8} \log | x - 1 | - \frac{1}{2 (x - 1)} + \frac{8}{5} \log | x + 3 | +$
D

none of these

Solution

$I = \int \frac{x^{2} + 1}{(x - 1)^{2} (x + 3)} d x$
$L e t \frac{x^{2} + 1}{(x - 1)^{2} (x + 3)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^{2}} + \frac{C}{x + 3} - - - - - (1)$
$x^{2} + 1 = A (x - 1) (x + 3) + B (x + 3) + C (x - 1)^{2} - - - - - - (2)$
$P u t t i n g x - 1 = 0, i . e ., x = 1 i n e q u a t i o n (2), w e g e t 2 = 4 B o r B = \frac{1}{2}$
$P u t t i n g x + 3 = 0, i . a ., x = - 3 i n e q u a t i o n (2), w e g e t 10 = 16 C o r C = \frac{5}{8}$
$E q u a t i n g t h e c o e f f i c i e n t s o f x^{2} o n b o t h t h e s i d e s o f t h e i d e n t i t y o f e q u a t i o n (2), w e g e t$
$1 = A + C o r A = 1 - C = 1 - \frac{5}{8} = \frac{3}{8}$
$s u b s t i t u t i n g t h e v a l u e s o f A, B i n e q u a t i o n (1), w e g e t$
$\frac{x^{2} + 1}{(x - 1)^{2} (x + 3)} = \frac{3}{8} \frac{1}{x - 1} + \frac{1}{2} \frac{1}{(x - 1)^{2}} + \frac{5}{8} (x + 3)$
$\begin{array}{l} I = \frac{3}{8} \int \frac{1}{x - 1} d x + \frac{1}{2} \int \frac{1}{(x - 1)^{2}} d x + \frac{5}{8} \int \frac{1}{x + 3} d x \\ = \frac{3}{8} \log | x - 1 | - \frac{1}{2 (x - 1)} + \frac{5}{8} \log | x + 3 | + C \end{array}$

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