(Solved): Find the Laplace transforms of the functions given in Problems 11 through 22. 11. \( f(t)=2 \) if \ ...

Find the Laplace transforms of the functions given in Problems 11 through 22. 11. \( f(t)=2 \) if \( 0 \leqq t<3 ; f(t)=0 \) if \( t \geqq 3 \) 12. \( f(t)=1 \) if \( 1 \leqq t \leqq 4 ; f(t)=0 \) if \( t<1 \) or if \( t>4 \) 13. \( f(t)=\sin t \) if \( 0 \leqq t \leqq 2 \pi ; f(t)=0 \) if \( t>2 \pi \) 14. \( f(t)=\cos \pi t \) if \( 0 \leqq t \leqq 2 ; f(t)=0 \) if \( t>2 \) 15. \( f(t)=\sin t \) if \( 0 \leqq t \leqq 3 \pi ; f(t)=0 \) if \( t>3 \pi \) 16. \( f(t)=\sin 2 t \) if \( \pi \leqq t \leqq 2 \pi ; f(t)=0 \) if \( t<\pi \) or if \( t>2 \pi \) 17. \( f(t)=\sin \pi t \) if \( 2 \leqq t \leqq 3 ; f(t)=0 \) if \( t<2 \) or if \( t>3 \) 18. \( f(t)=\cos \frac{1}{2} \pi t \) if \( 3 \leqq t \leqq 5 ; f(t)=0 \) if \( t<3 \) or if \( t>5 \) 19. \( f(t)=0 \) if \( t<1 ; f(t)=t \) if \( t \geqq 1 \) 20. \( f(t)=t \) if \( t \leqq 1 ; f(t)=1 \) if \( t>1 \) 21. \( f(t)=t \) if \( t \leqq 1 ; f(t)=2-t \) if \( 1 \leqq t \leqq 2 \); \( f(t)=0 \) if \( t>2 \) 22. \( f(t)=t^{3} \) if \( 1 \leqq t \leqq 2 ; f(t)=0 \) if \( t<1 \) or if \( t>2 \)