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Please Solve R.D.Sharma class 12 Chapter 19 Definite Integrals Exercise 19.1 Question 4 Maths textbook Solution.

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Answer: \frac{\pi }{4}

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

Given: \int_{0}^{1}\frac{1}{1+x^{2}}dx

Solution: \int_{0}^{1}\frac{1}{1+x^{2}}dx=\int \tan ^{-1}xdx                            \left [ \int \frac{1}{1+x^{2}}dx=\tan ^{-1}x \right ]

=\tan ^{-1}(1)-\tan ^{-1}(0) \quad\left[\tan \left(\frac{\pi}{4}\right)=1 \Rightarrow \tan ^{-1}(1)=\frac{\pi}{4}, \tan 0=0 \Rightarrow \tan ^{-1}(0)=0\right]

=\left [ \frac{\pi }{4}-0 \right ]=\frac{\pi }{4}

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