How can I calculate this integral using the fundamental theorem of calculus? [closed]
Can someone please explain how to calculate this integral. I'm pretty sure you have to use the first part of the fundamental theorem of calculus but I can't figure out how: $$\frac{d}{dx} \int_0^{\pi/2} \sin\frac{x}{2} \cos\frac{x}{3} \,dx $$
Solution 1:
$\sin\frac x2\cos\frac x3$ is continuous and hence integrable on $[0,\frac{\pi}2]$, so the integral $$\int_0^{\frac{\pi}2}\sin\frac x2\cos\frac x3dx$$ exists.
Let the value of the integral be $I$. The whole expression becomes $$\frac{d}{dx}I$$ which obviously evaluates to zero.