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Jonathan Kent

Jonathan Kent

Answered question

2022-05-22

How to prove that 0 cos ( α t ) e λ t d t = λ α 2 + λ 2 using real methods?

Answer & Explanation

Mya Hurst

Mya Hurst

Beginner2022-05-23Added 13 answers

Explicitly, with the choice
u = cos α t , d u = α sin α t d t , d v = e λ t , v = λ 1 e λ t ,
the first integration by parts gives
I = e λ t cos α t d t = 1 λ e λ t cos α t α λ e λ t sin α t .
Then, the second integration by parts with the choice
u = sin α t , d u = α cos α t d t , d v = e λ t , v = λ 1 e λ t ,
gives
I = 1 λ e λ t cos α t + α λ 2 e λ t sin α t α 2 λ 2 e λ t cos α t d t .
But this integral on the RHS is simply I, so we conclude
( 1 + α 2 λ 2 ) I = e λ t λ 2 ( α sin α t λ cos α t ) ,
which gives the value of the indefinite integral as
I = e λ t λ 2 + α 2 ( α sin α t λ cos α t ) + C ,
for some constant of integration C. Taking the limit as t , noting that sinαt and cosαt are bounded and finite, gives
t = 0 e λ t cos α t d t = I ( ) I ( 0 ) = 0 ( λ λ 2 + α 2 ) = λ λ 2 + α 2 ,
as claimed.

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