calculate-f-x-0-e-x-t-sin-t-dt-with-x-gt-0-2-calculate-0-e-3-t-sin-t-dt-

Question Number 59282 by Mr X pcx last updated on 07/May/19

c a l c u l a t e f (x) = \int_{0}^{\infty} e^{- x [t]} s i n ([t]) d t

w i t h x > 0

2) c a l c u l a t e \int_{0}^{\infty} e^{- 3 [t]} s i n ([t]) d t .

Commented by maxmathsup by imad last updated on 08/May/19

1) w e h a v e f (x) = \sum_{n = 0}^{\infty} \int_{n}^{n + 1} e^{- n x} s i n (n) d t = \sum_{n = 0}^{\infty} s i n (n) e^{- n x} (n + 1 - n)

= \sum_{n = 0}^{\infty} e^{- n x} s i n (n) = I m (\sum_{n = 0}^{\infty} e^{- n x + i n}) b u t

\sum_{n = 0}^{\infty} e^{- n x + i n} = \sum_{n = 0}^{\infty} {(e^{i - x})}^{n} = \frac{1}{1 - e^{i - x}} = \frac{1}{1 - e^{- x} (c o s (1) + i s i n (1)}

= \frac{1}{1 - e^{- x} c o s (1) - i e^{- x} s i n (1)} = \frac{1 - e^{- x} c o s (1) + {i e}^{- x} s i n (1)}{{(1 - e^{- x} c o s (1))}^{2} + e^{- 2 x} {s i n}^{2} (1)} \Rightarrow

f (x) = \frac{e^{- x} s i n (1)}{{(1 - e^{- x} c o s (1))}^{2} + e^{- 2 x} {s i n}^{2} (1)} = \frac{e^{- x} s i n (1)}{1 - 2 e^{- x} c o s (1) + e^{- 2 x}}

2) \int_{0}^{\infty} e^{- 3 [t]} s i n ([t]) d t = f (3) = \frac{e^{- 3} s i n (1)}{e^{- 6} - 2 e^{- 3} c o s (1) + 1} = \frac{e^{3} s i n (1)}{1 - 2 e^{3} c o s (1) + e^{6}} .

Answered by tanmay last updated on 07/May/19

f (x) = \int_{0}^{1} e^{- x \times 0} s i n (0) d t + \int_{1}^{2} e^{- x \times 1} s i n (1) d t +

\int_{2}^{3} e^{- x \times 2} s i n (2) d t + \dots + \int_{r}^{r + 1} e^{- x \times r} s i n (r) d t + \dots

= 0 + e^{- x} s i n 1 + e^{- 2 x} s i n 2 + e^{- 3 x} s i n 3 + \dots

= \underset{r = 0}{\sum^{\infty}} e^{- r x} \times s i n r

Q = e^{- x} s i n 1 + e^{- 2 x} s i n 2 + e^{- 3 x} s i n 3 + \dots

P = e^{- x} c o s 1 + e^{- 2 x} c o s 2 + e^{- 3 x} c o s 3 + \dots

P + i Q = e^{- x} \times e^{i} + e^{- 2 x} \times e^{i 2} + e^{- 3 x} \times e^{i 3} + \dots

S = t + t^{2} + t^{3} + \dots \infty [t = \frac{e^{i}}{e^{x}}]

S = \frac{t}{1 - t} = \frac{e^{i}}{e^{x} (1 - \frac{e^{i}}{e^{x}})} = \frac{c o s 1 + i s i n 1}{e^{x} - c o s 1 - i s i n 1}

S = \frac{(c o s 1 + i s i n 1)}{{(e^{x} - c o s 1)}^{2} + {s i n}^{2} 1} \times (e^{x} - c o s 1 + i s i n 1)

S = \frac{c o s 1 (e^{x} - c o s 1) + i s i n 1 c o s 1 + i s i n 1 (e^{x} - c o s 1) - {s i n}^{2} 1}{e^{2 x} - 2 e^{x} c o s 1 + 1}

= \frac{e^{x} c o s 1 - 1 + i (s i n 1 c o s 1 + e^{x} s i n 1 - s i n 1 c o s 1)}{e^{2 x} - 2 e^{x} c o s 1 + 1}

s o r e a u i r e d a n s i s Q = (c o m p l e x p a r t)

= \frac{e^{x} s i n 1}{e^{2 x} - 2 e^{x} c o s 1 + 1}

p l s c h e c k \dots

2) \frac{e^{3} s i n 1}{e^{6} - 2 e^{3} c o s 1 + 1}

Commented by maxmathsup by imad last updated on 08/May/19

s i r T a n m a y y o u r a n s w e r i s c o r r e c t t h a n k s \dots

Commented by tanmay last updated on 08/May/19

m o s t w e l c o m e s i r

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