f ′ ( x ) = ∫ 0 1 2 cos − 1 ( x ) ⋅ d x {\displaystyle f'(x)=\int _{0}^{\frac {1}{2}}\cos ^{-1}(x)\cdot dx} ∫ 0 1 2 cos − 1 ( x ) ⋅ d x {\displaystyle \int _{0}^{\frac {1}{2}}\cos ^{-1}(x)\cdot dx} = ∫ 0 π e c o s t ( 2 s i n t c o s t ) d t {\displaystyle \int _{0}^{\pi }e^{cost}(2sintcost)dt} = − 2 ∫ c o s 0 c o s π e u u d u {\displaystyle {-2}\int _{cos0}^{cos\pi }e^{u}{u}du} = − 2 u e u | c o s 0 c o s π − ( − 2 ) ∫ c o s 0 c o s π e u d u {\displaystyle {-2}{u}e^{u}{\bigg |}_{cos0}^{cos\pi }-(-2)\int _{cos0}^{cos\pi }e^{u}du}
= − 2 u e u | c o s 0 c o s π + 2 e u | c o s 0 c o s π d u {\displaystyle {-2}{u}e^{u}{\bigg |}_{cos0}^{cos\pi }+{2}e^{u}{\bigg |}_{cos0}^{cos\pi }du} = 2 u e u | c o s π c o s 0 − 2 e u | c o s π c o s 0 d u {\displaystyle {2}{u}e^{u}{\bigg |}_{cos\pi }^{cos0}-{2}e^{u}{\bigg |}_{cos\pi }^{cos0}du} = 2 c o s ( 0 ) e c o s ( 0 ) − 2 c o s ( π ) e c o s ( π ) − 2 e c o s ( 0 ) + 2 e c o s ( π ) {\displaystyle {2}{cos(0)}e^{cos(0)}-{2}{cos(\pi )}e^{cos(\pi )}-{2}e^{cos(0)}+{2}e^{cos(\pi )}}
= 2 ( 1 ) e 1 − 2 ( − 1 ) e − 1 − 2 e 1 + 2 e − 1 {\displaystyle {2}(1)e^{1}-{2}(-1)e^{-1}-{2}e^{1}+{2}e^{-1}} = 2 e 1 + 2 e − 1 − 2 e 1 + 2 e − 1 {\displaystyle {2}e^{1}+{2}e^{-1}-{2}e^{1}+{2}e^{-1}} = 2 e − 1 + 2 e − 1 {\displaystyle {2}e^{-1}+{2}e^{-1}} = 4 e − 1 {\displaystyle {4}e^{-1}}
