Question

Evaluate the definite integral ${\int }_0^{\frac{x}{4}}\sin 2xdx$

Answer 1

Let $I={\int }_0^{\frac{x}{4}}\sin 2xdx$

$\Rightarrow \int \sin 2xdx=\left(\frac{-\cos 2x}{2}\right)=F\left(x\right)$

By second fundamental theorem of calculus, we obtain

$I=F\left(\frac{\pi }{4}\right)-F\left(0\right)$

$=-\frac{1}{2}[\cos 2\left(\frac{\pi }{4}\right)-\cos 0]$

$=-\frac{1}{2}[\cos \left(\frac{\pi }{2}\right)-\cos 0]$

$=-\frac{1}{2}[0-1]=\frac{1}{2}$