Question

The value of $\underset{2}{\overset{3}{\int }}\frac{xdx}{x^2+1}$ is:

• A. $\frac{1}{2}\ln 3$
  
• B. $\ln 2$
• C. $\frac{1}{2}\ln 2$
• D. $\ln 5$

Answer 1

The correct option is C $\frac{1}{2}\ln 2$

Let $I=\underset{2}{\overset{3}{\int }}\frac{xdx}{x^2+1}$
Put $x^2+1=t$
So,
$d(x^2+1)=dt$
$\Rightarrow 2xdx=dt\Rightarrow xdx=\frac{dt}{2}$
When $x=2$; $t=2^2+1=5$
When $x=3$; $t=3^2+1=10$
$I=\underset{5}{\overset{10}{\int }}\frac{dt}{2t}=\frac{1}{2}{\int }_5^{10}\frac{dt}{t}$
[We know that$\int \frac{1}{x}dx=\ln |x|$]
$=\frac{1}{2}{\left[\ln t\right]}_5^{10}=\frac{1}{2}(\ln 10-\ln 5)$
$\Rightarrow I=\frac{1}{2}\ln 2$