The value of [_ - 2^ 2 ( 1 + ^2x 1 + ^ x ) ,dx] is

Question

The value of $$\int\limits_{ - {\pi  \over 2}}^{{\pi  \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx$$ is

Accepted Answer

$$I = \int\limits_0^{{\pi  \over 2}} {{{(1 + {{\sin }^2}x)} \over {(1 + {\pi ^{\sin x}})}} + {{{\pi ^{\sin x}}(1 + {{\sin }^2}x)} \over {(1 + {\pi ^{\sin x}})}}} dx$$

$$I = \int_0^{{\pi  \over 2}} {(1 + {{\sin }^2}x)\,dx} $$

$$I = {\pi  \over 2} + {\pi  \over 2}.{1 \over 2} = {{3\pi } \over 4}$$

Solutions · 1