The integral $16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}}$ is equal to
Solution
$I=\int \frac{d x}{x^{3}\left(x^{2}+2\right)^{2}}$ <br/><br/>$=\frac{1}{4} \int \frac{x}{x^{2}+2} d x+\frac{1}{4} \int \frac{x}{\left(x^{2}+2\right)^{2}}-\frac{1}{4} \int \frac{d x}{x}+\frac{1}{4} \int \frac{d x}{x^{3}}$<br/><br/> $=\frac{1}{8} \ln \left(x^{2}+2\right)-\frac{\ln x}{4}-\frac{1}{8\left(x^{2}+2\right)}-\frac{1}{8 x^{3}}$
<br/><br/>
Now, $16 \int_{1}^{2} \frac{d x}{x^{3}\left(x^{2}+2\right)^{2}}=2 \ln 6-2 \ln 3-4 \ln 2+\frac{11}{6}$ <br/><br/>$=\frac{11}{6}-\ln 4$
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
This question is part of PrepWiser's free JEE Main question bank. 216 more solved questions on Definite Integration are available — start with the harder ones if your accuracy is >70%.