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Provide solution for  RD Sharma maths Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 32 Subquestion (ii) textbook solution.

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Answer:- 0

Hints:-  You must know the integration rules of trignometric functions .

Given:-  \int_{0}^{1} \tan ^{-1}\left(\frac{1-2 x}{1+x-x^{2}}\right) d x

Solution :

\begin{aligned} &I=\int_{0}^{1} \tan ^{-1}\left(\frac{1-2 x}{1+x-x^{2}}\right) d x \\ &I=\int_{0}^{1} \tan ^{-1} \frac{-x+(1-x)}{1+x(1-x)} d x \\ &I=\int_{0}^{1} \tan ^{-1}(-x)+\tan ^{-1}(1-x) d x \end{aligned}                                     ....(1)

\begin{aligned} &I=\int_{0}^{1} \tan ^{-1}(1+x) d x+\tan ^{-1}(1-1+x) d x \\ &I=\int_{0}^{1} \tan ^{-1}(1-x)+\tan ^{-1}(x) d x \end{aligned}                  .....(2)

Adding both

 \begin{aligned} &2 I=\int_{0}^{1}\left(\tan ^{-1}(x)+\tan ^{-1}(1-x)-\tan ^{-1}(1-x)-\tan ^{-1}(x)\right) d x \\ &2 I=0 \\ &I=0 \end{aligned}

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