5.3 The Fundamental Theorem of Calculus/41

${\begin{aligned}\int \limits _{0}^{\pi }f(x)dx\quad {\text{where}}\;f(x)={\begin{cases}sin(x)&0\leq x<{\frac {\pi }{2}}\\cos(x)&{\frac {\pi }{2}}\leq x\leq \pi \end{cases}}&=\int \limits _{0}^{\frac {\pi }{2}}f(x)dx+\int \limits _{\frac {\pi }{2}}^{\pi }f(x)dx=\int \limits _{0}^{\frac {\pi }{2}}\sin(x)dx+\int \limits _{\frac {\pi }{2}}^{\pi }\cos(x)dx\\[2ex]&=-\cos(x)\\[2ex]\end{aligned}}$