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#### Please Solve R.D.Sharma class 12 Chapter 19 Definite Integrals Exercise 19.1 Question 5 Maths textbook Solution.

Answer: $log\left ( \sqrt{2} \right )$

Hint:Use indefinite formula to solve the integral and then put the value of limit to get the required answer.

Given: $\int_{2}^{3}\frac{x}{x^{2}+1}dx$

Solution:$\int_{2}^{3}\frac{x}{x^{2}+1}dx$

Put $x^{2}+1=t\Rightarrow 2xdx=dt\Rightarrow xdx=\frac{dt}{2}$

When $x=2$ then$t=3^{2}+1=4+1=5$

When $x=3$ then $t=3^{2}+1=9+1=10$

Then$\int_{2}^{3}\frac{x}{x^{2}+1}dx=\frac{1}{2}\int_{5}^{10}\frac{1}{t}dt=\frac{1}{2}\left [ log|t| \right ]^{10}_{5}$                                            $\left [ \int \frac{1}{x}dx=log|x| \right ]$

$=\frac{1}{2}\left [ log10-log5 \right ]$                                                                $\left [ log\; \; a -log\; b=log\frac{a}{b}\right ]$

$=\frac{1}{2}log\frac{10}{5}$

$=\frac{1}{2}log2$                                                                                    $\left [ log\: a^{m}=m\: log\: a \right ]$

$=log\left ( 2^{\frac{1}{2}} \right )$

$=log\sqrt{2}$