∫ 0 π f ( x ) d x where f ( x ) = { sin ( x ) 0 ≤ x < π 2 cos ( x ) π 2 ≤ x ≤ π {\displaystyle \int _{0}^{\pi }f(x)\,dx\quad {\text{where}}\;f(x)={\begin{cases}\sin(x)&0\leq x<{\frac {\pi }{2}}\\\cos(x)&{\frac {\pi }{2}}\leq x\leq \pi \end{cases}}}
= ∫ 0 π 2 f ( x ) d x + ∫ π 2 π f ( x ) d x = ∫ 0 π 2 sin ( x ) d x + ∫ π 2 π cos ( x ) d x {\displaystyle =\int \limits _{0}^{\frac {\pi }{2}}f(x)dx+\int \limits _{\frac {\pi }{2}}^{\pi }f(x)dx=\int \limits _{0}^{\frac {\pi }{2}}\sin(x)dx+\int \limits _{\frac {\pi }{2}}^{\pi }\cos(x)dx}
= − cos ( x ) | 0 π 2 + sin ( x ) | π 2 π = [ − cos ( π 2 ) + cos ( 0 ) ] + [ sin ( π ) − sin ( π 2 ) ] {\displaystyle =-\cos(x){\bigg |}_{0}^{\frac {\pi }{2}}+\sin(x){\bigg |}_{\frac {\pi }{2}}^{\pi }=\left[-\cos({\frac {\pi }{2}})+\cos(0)\right]+\left[\sin(\pi )-\sin({\frac {\pi }{2}})\right]}
= 0 + 1 + 0 − 1 = 0 {\displaystyle =0+1+0-1=0}
![{\displaystyle =-\cos(x){\bigg |}_{0}^{\frac {\pi }{2}}+\sin(x){\bigg |}_{\frac {\pi }{2}}^{\pi }=\left[-\cos({\frac {\pi }{2}})+\cos(0)\right]+\left[\sin(\pi )-\sin({\frac {\pi }{2}})\right]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5b9732d807a25b11d42bb8e37edf9d0798775994)
