\(\int \frac{dx}{(x^2-9)}\) equals ______

Category: Questions\(\int \frac{dx}{(x^2-9)}\) equals ______
Editor">Editor Staff asked 11 months ago

\int \frac{dx}{(x^2-9)} equals ______
 
(a) \frac{1}{6} log \frac{x+3}{x-3} + C
 
(b) \frac{1}{6} log \frac{x-3}{x+3} + C
 
(c) \frac{1}{5} log \frac{x+3}{x-3} + C
 
(d) \frac{1}{3} log \frac{x+3}{x-3} + C
 
This question was addressed to me in an interview for job.
 
My question is from Integration by Partial Fractions in section Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

The correct answer is (b) \frac{1}{6} log \frac{x-3}{x+3} + C
 
The explanation is: \int \frac{dx}{(x^2-9)}=\frac{A}{(x-3)} + \frac{B}{(x+3)}
 
By simplifying, it we get \frac{A(x+3)+B(x-3)}{(x^2-9)} = \frac{(A+B)x+3A-3B}{(x^2-9)}
 
By solving the equations, we get, A+B=0 and 3A-3B=1
 
By solving these 2 equations, we get values of A=1/6 and B=-1/6.
 
Now by putting values in the equation and integrating it we get value,
 
\frac{1}{6} log (\frac{x-3}{x+3}) + C.