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# Integrate: (e^s)sin(t-s)ds?

Please show me how you got your answers.

Also: The upperlimit of the integral is the variable s and the lowerlimit is 0.

Thank you.

### 1 Answer

- AlexanderLv 61 decade agoFavorite Answer
I(t,S) =

Int of exp(s)sin(t-s) ds =

Int of sin(t-s) d exp(s) =

exp(s)sin(t-s) - Int of exp(s) d sin(t-s) =

exp(s)sin(t-s) + Int of exp(s) cos(t-s) ds =

exp(s)sin(t-s) + Int of cos(t-s) d exp(s) =

exp(s)sin(t-s) + cos(t-s) exp(s) -

Int of exp(s) d cos(t-s) =

exp(s)sin(t-s) + cos(t-s) exp(s) -

Int of exp(s) sin(t-s) ds + C =

exp(s)sin(t-s) + cos(t-s) exp(s) - I(t,S) + C

***************** Now:

I(t,S) = exp(s)sin(t-s) + cos(t-s) exp(s) - I(t,S) + C

2I(t,S) = exp(s)sin(t-s) + cos(t-s) exp(s) + C

I(t,S) = 1/2 exp(s)[sin(t-s) + cos(t-s)] + C

I(t,S) = 1/√2 exp(s) sin(t-s+π/4) + C

and

I(t,0) = 1/√2 sin(t+π/4) + C = 0

Answer:

I(t,S) = 1/√2 [exp(s) sin(t-s+π/4) - sin(t+π/4)]