Question

The value of definite integral $\underset{0}{\overset{1}{\int }}\frac{x^2+5}{(x+1{)}^2(x-2)}dx$ is equal to

• A. $\ln 2$
• B. $\ln 2-1$
• C. $-(1+\ln 2)$
• D. $-\ln 2$

Answer 1

Answer: (C)

$I=\underset{0}{\overset{1}{\int }}\frac{\left(x^2+2x+1\right)-2(x-2)}{(x+1{)}^2(x-2)}dx$
${{I=\underset{0}{\overset{1}{\int }}\frac{dx}{x-2}-2{\int }_0^1\frac{dx}{(x+1{)}^2}\Rightarrow I=\ln |x-2|]}_0^1+\frac{2}{x+1}]}_0^1=(0-\ln 2)+(1-2)$
$I=-(1+\ln 2).$