5.3 The Fundamental Theorem of Calculus/41

$\int _{0}^{\pi }f(x)\,dx\quad {\text{where}}\;f(x)={\begin{cases}\sin(x)&0\leq x<{\frac {\pi }{2}}\\\cos(x)&{\frac {\pi }{2}}\leq x\leq \pi \end{cases}}$

$=\int \limits _{0}^{\frac {\pi }{2}}f(x)dx+\int \limits _{\frac {\pi }{2}}^{\pi }f(x)dx=\int \limits _{0}^{\frac {\pi }{2}}\sin(x)dx+\int \limits _{\frac {\pi }{2}}^{\pi }\cos(x)dx$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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] }

$=0+1+0-1=0$