nice-calculus-calculate-F-0-te-t-sin-3-t-dt-

Question Number 139734 by mnjuly1970 last updated on 30/Apr/21

\dots . . n i c e \dots c a l c u l u s \dots . .

c a l c u l a t e ::

F := \int_{0}^{\infty} {t e}^{- t} {s i n}^{3} (t) d t = ?

\dots \dots \dots \dots \dots \dots \dots . .

Answered by Dwaipayan Shikari last updated on 30/Apr/21

\int_{0}^{\infty} {t e}^{- t} (\frac{3}{4} s i n (t) - \frac{1}{4} s i n (3 t)) d t

= \frac{3}{8 i} \int_{0}^{\infty} {t e}^{- t + i t} - {t e}^{- t (1 + i)} - \frac{1}{8 i} \int_{0}^{\infty} {t e}^{- t + 3 i t} - {t e}^{- t (1 + 3 i)} d t

= \frac{3}{8 i} . (\frac{1}{{(1 - i)}^{2}} - \frac{1}{{(1 + i)}^{2}}) - \frac{1}{8 i} (\frac{1}{{(1 - 3 i)}^{2}} - \frac{1}{{(1 + 3 i)}^{2}})

= \frac{3}{8 i} (\frac{4 i}{{(1 + 1)}^{2}}) - \frac{1}{8 i} (\frac{12 i}{{(10)}^{2}}) = \frac{3}{8} - \frac{3}{200} = \frac{72}{200} = \frac{9}{25}

Commented by mnjuly1970 last updated on 30/Apr/21

t h a n k s a l o t \dots