Question

Find the value of \(\int_0^{\infty} tsin(t)cos(t)\).

(a) s ⁄ s^2+2^2

(b) a ⁄ a^2+s^4

(c) 1

(d) 0

This question was posed to me during an interview.

My question comes from Laplace Transform by Properties topic in division Laplace Transform of Engineering Mathematics

Answer 1

Correct answer is (d) 0

The explanation is: y(t)=\(\frac{1}{2} t Sin(2t)u(t)\)

Laplace transform of Sin(2t)=\(\frac{2}{s^2+4}\)

Laplace transform of tSin(2t)=\(-\frac{d}{dt} \frac{2}{s^2+4}=\frac{2(2s)}{(s^2+4)^2}=\frac{4s}{(s^2+4)^2}\)

Laplace transform of \(\frac{1}{2}tsin(2t)=\int_{-0}^{\infty} e^{-st} tsin(t)cos(t)dt=\frac{2s}{[s^2+4]^2}\)

Putting, s = 0, \(\int_0^{\infty} tsin(t)cos(t)dt=0\)