Question

Evaluate $\int \frac{5x}{(x+1)(x^2-4)}dx$.

• A. $5(\frac{-ln|x+2|}{2}+\frac{ln|x+1|}{3}+\frac{ln|x-2|}{6})$
• B. $\frac{-ln|x+2|}{2}+\frac{ln|x+1|}{3}+\frac{ln|x-2|}{6}$
• C. $5(\frac{-ln|x+2|}{2}-\frac{ln|x+1|}{3}-\frac{ln|x-2|}{6})$
• D. $5(\frac{-ln|x+2|}{2}+\frac{ln|x+1|}{2}+\frac{ln|x-2|}{3})$

Answer 1

Answer: (A)

$5(\frac{-ln|x+2|}{2}+\frac{ln|x+1|}{3}+\frac{ln|x-2|}{6})$

$\int \frac{5x}{(x+1)(x^2-4)}dx=5\int \frac{A}{x+1}+\frac{B}{x-2}+\frac{C}{x+2}$

$1=A(x-2)(x+2)+B(x+2)(x+1)+C(x-2)(x+1)$

Now, Comparing the Cofficients of $x^3,x^2,x$ and constants

We get,

$A=\frac{1}{3},B=\frac{1}{6},C=\frac{-1}{2}$

Hence,

$5\int \frac{1}{3}\frac{1}{x+1}+\frac{1}{6}\frac{1}{x-2}-\frac{1}{2}\frac{1}{x+2}=5(\frac{\ln |x+1|}{3}+\frac{\ln (x-2)}{6}-\frac{\ln (x+2)}{2})$