L(sint)=1s2+1
By "first shifting property" of Laplace transformation:
L(e−3tsint)=1(s+3)2+1
L(e−3tsint)=1(s2+6s+9)+1
L(e−3tsint)=1s2+6s+10
From "multiplication by power of t":
L(te−3tsint)=(−1)1dds(1s2+6s+10)
L(te−3tsint)=− −1(2s+6)(s2+6s+10)2
L(te−3tsint)=2s+6(s2+6s+10)2
Therefore,
∫∞0te−3tsint dt=lim
\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt = \dfrac{2(0) + 6}{[ \, 0^2 + 6(0) + 10 \, ]^2}
\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt = \dfrac{6}{10^2}
\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt = \dfrac{6}{100}
\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt = \dfrac{3}{50} answer