∫ 0 π f ( x ) d x where f ( x ) = { sin ( x ) 0 ≤ x < π 2 cos ( x ) π 2 ≤ x ≤ π {\displaystyle \int \limits _{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 = − cos ( x ) | 0 π 2 + sin ( x ) | π 2 π {\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){\bigg |}_{0}^{\frac {\pi }{2}}+\sin(x){\bigg |}_{\frac {\pi }{2}}^{\pi }}
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= -\cos(\frac{\pi}{2}) + \cos(0)
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