Question

Statement-l ${\int }_0^{\pi /2}\frac{dx}{1+{\tan }^{5}x}=\frac{\pi }{4}$
Statement 2:${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a+x)dx={\int }_0^{\pi /2}\frac{dx}{1+{\tan }^{3}x}={\int }_0^{\pi /2}\frac{d_X}{1+{\cot }^{3}x}=\frac{\pi }{4}$

• A. Statement 1 is True, Statement 2 is True; Statement 2 is a correct exlanation for Statement 1
• B. Statement 1 is True, Statement 2 is True; Statement 2 Not a correct exlanation for Statement 1
• C. Statement 1 is True, Statement 2 is False
• D. Statement 1 is False, Statement 2 is True

Answer 1

The correct option is C Statement 1 is True, Statement 2 is False

Statement 1 :

${\int }_0^{\frac{\pi }{2}}\frac{dx}{1+{\tan }^{5}x}=\frac{\pi }{4}\Rightarrow LHS={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{5}x}{{\cos }^{5}x+{\sin }^{5}x}dx\Rightarrow {\int }_0^{\frac{\pi }{2}}\frac{f\left(\cos x\right)}{f\left(\cos x\right)+f\left(\sin x\right)}dx\Rightarrow \frac{\pi }{4}=RHS$

Statement 2 :

${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(x+a)dx\Rightarrow {\int }_0^{\frac{\pi }{2}}\frac{dx}{1+{\tan }^{3}x}\Rightarrow {\int }_0^{\frac{\pi }{2}}\frac{dx}{1+{\cot }^{3}x}\Rightarrow \frac{\pi }{4}$

But ${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$

Hence, statement 2 is false.