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

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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


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

When x=2 thent=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}log2                                                                                    \left [ log\: a^{m}=m\: log\: a \right ]

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


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