Correct Answer - Option 3 : 0
Concept:
Odd function: f(-x) = -f(x)
Even the function: f(-x) = f(x)
sin (-x) = -sin x
cos (-x) = cos x
tan (-x) = -tan x
Property of definte integral:
\(\rm\int_{-a}^{a}f(x)dx=\left\{\begin{matrix} \rm2\int_{0}^{a} f(x)dx&, \rm \text{ Even function }\\ 0&, \text{Odd function} \end{matrix}\right.\)
Calculation:
\(\rm f(x)=sinx+tanx\)
\(\rm f(-x)=sin(-x)+tan(-x)\)
\(\rm f(-x)=-(sin(x)+tan(x))\)
Hence f(x) is odd function
\(\rm \therefore \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}(\sin x+\tan x)dx=0\)
Hence , option 3 is correct.
