Question

need help on the attachment

Answer 1

Break it up:\[5\int\limits_{0}^{\pi/4}\cos (2x) dx -3\int\limits_{0}^{\pi/4}\sin (2x)dx\]
u-substitue:
Let u=2x so du=2dx and (1/2)du = dx

I like to use new u limits from my substitution:
If x=0, u= 2(0)=0; If x=pi/4, u= 2(pi/4)=(pi/2)

The u integral becomes:
\[(5/2)\int\limits_{0}^{\pi/2}\cos (u)du-(3/2)\int\limits_{0}^{\pi/2}\sin(u)du\]
Take the anti-derivatives:
\[(5/2)\left[ \sin u \right]_{0}^{\pi/2}-(3/2)\left[ -\cos u \right]_{0}^{\pi/2}\]
\[(5/2)\left[ \sin (\pi/2)-\sin (0) \right]-(3/2)\left[ -\cos (\pi/2)+\cos (0) \right]\]
(5/2)-(3/2)=1