Menu Close

let-R-and-a-gt-0-find-0-e-ax-cos-x-dx-

Tinku Tara 0 Comments
Pin




Question Number 31053 by abdo imad last updated on 02/Mar/18
let λ ∈R and a>0 find ∫_0 ^∞ e^(−ax) cos(λx)dx .
letλ∈Randa>0find∫0∞e−axcos(λx)dx.
Commented by abdo imad last updated on 03/Mar/18
let put I=∫_0 ^∞ e^(−ax) cos(λx)dx I=Re( ∫_0 ^∞ e^(−ax) e^(iλx) dx)=Re( ∫_0 ^∞ e^((−a+iλ)x) dx)but ∫_0 ^∞ e^((−a+iλ)x) dx=[ (1/(−a+iλ))e^((−a+iλ)x) ]_(x=0) ^(x→+∞) =((−1)/(−a+iλ))= (1/(a−λi)) =((a+λi)/(a^2 +λ^2 )) ⇒ I= (a/(a^2 +λ^2 )) .
letputI=∫0∞e−axcos(λx)dxI=Re(∫0∞e−axeiλxdx)=Re(∫0∞e(−a+iλ)xdx)but∫0∞e(−a+iλ)xdx=[1−a+iλe(−a+iλ)x]x=0x→+∞=−1−a+iλ=1a−λi=a+λia2+λ2⇒I=aa2+λ2.

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *