Q:

Trigonometric Functions. Definite Integrals

Find, correct to one decimal place, the value of 
$$\int_{0}^{60} 2\sin(x/2) \, dx.$$
Can someone please show me how this question is done. It would be very helpful thanks!

A:

To solve this integral one must use a substitution. If we set $u=x/2$, then $du=(1/2)dx$, which gives $2\,du=dx$. Thus, the integral then becomes, $$2\int_0^{30} 2\sin(u)\,du.$$ To get the limits on the integral, we used our substitution again, that is, at $x=0$, $u=0/2=0$ and at $x=60$, $u=60/2=30$. With all the pieces determined, the integral evaluates to,
$$2\int_0^{30}2\sin(u)du=2(-2\cos(u))\bigg|_0^{30}=-4(\cos(30)-1)$$

