Answer
a) $\sin \pi x+C$
b) $\sin \dfrac{\pi x}{2}+C$
c) $\dfrac{2}{\pi} \sin \dfrac{\pi x}{2}+\pi \sin x+C$
Work Step by Step
a) The anti-derivative is:
Thus, $(\pi) (\dfrac{1}{\pi} \sin \pi x)+C=\sin \pi x+C$
b) The anti-derivative is:
Thus, we have $(\dfrac{\pi}{2})(\dfrac{1}{\pi/2}) \sin (\dfrac{\pi x}{2})+C=\sin (\dfrac{\pi x}{2})+C$
c) The anti-derivative is:
Thus, we have $(\dfrac{1}{\pi/2}) \sin (\dfrac{\pi x}{2})+\pi \sin x+C=(\dfrac{2}{\pi}) \sin (\dfrac{\pi x}{2})+\pi \sin x+C$
