Statement 1: ∫-33 dx1+10x1+x2=π3Statement 2:∫−aa f(x)dx=2∫0a f(x)dx

∫-33 dx1+10x1+x2 =∫03 11+10x1+x2+ 11+10-x1+x2dx =∫0311+x2dx=tan-1x03=π3 ∫−aa f(x)dx=2∫0a f(x)dx if fx is even 0 if fx is odd

Statement 1: ∫-33 dx1+10x1+x2=π3Statement 2:∫−aa f(x)dx=2∫0a f(x)dx

∫-33 dx1+10x1+x2 =∫03 11+10x1+x2+ 11+10-x1+x2dx =∫0311+x2dx=tan-1x03=π3 ∫−aa f(x)dx=2∫0a f(x)dx if fx is even 0 if fx is odd

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Statement 1: $\int_{- \sqrt{3}}^{\sqrt{3}} \frac{dx}{(1 + 10^{x}) (1 + x^{2})} = \frac{π}{3}$
Statement 2: $\int_{- a}^{a} f (x) dx = 2 \int_{0}^{a} f (x) dx$

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Statement 1 is True, Statement 2 is True, Statement ‘2’ is correct explanation for Statement 2

Statement 1 is True, Statement 2 is False

Statement 1 is False, Statement 2 is true

Statement 1 is True, Statement 2 is True, Statement ‘2’ is not correct explanation for Statement 2

answer is C.

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Detailed Solution

$\int_{- \sqrt{3}}^{\sqrt{3}} \frac{dx}{(1 + 10^{x}) (1 + x^{2})} = \int_{0}^{\sqrt{3}} [\frac{1}{(1 + 10^{x}) (1 + x^{2})} + \frac{1}{(1 + 10^{- x}) (1 + x^{2})}] d x = \int_{0}^{\sqrt{3}} (\frac{1}{(1 + x^{2})}) d x = {(\tan^{- 1} x)}_{0}^{\sqrt{3}} = \frac{π}{3} \int_{- a}^{a} f (x) dx = \{\begin{cases} 2 \int_{0}^{a} f (x) dx i f f (x) i s e v e n \\ 0 i f f (x) i s o d d \end{cases}$