Find \(\rm \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left ( \sin x+\tan x \right ) dx\)

Question

Question

Answer 1

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.