NIMCET 2015 Question Paper with solutions

Question : 4 of 120

Marks: +1, -0

If ∫

e^x

(f (x) - f ' (x)) dx = Φ (x) , then the value of ∫

e^x

f (x) dx is :

Solution:

We have

∫e^{x}[f(x)-f'(x)]dx=ϕx

⇒∫e^{x}[f(x)dx-∫e^{x}f'(x)dx=ϕ(x)\.

⇒∫e^{x}f(x)dx-[e^{x}∫f'(x)dx-∫{de^{x}}/{dx}∫f(x)dx\,⋅dx]=ϕx

⇒∫e^{x}f(x)dx-[e^{x}f(x)-∫e^{x}f'(x)dx]=ϕx

⇒∫e^{x}f(x)dx-e^{x}f(x)+∫e^{x}f(x)dx=ϕx

⇒2∫e^{x}f(x)dx=ϕ(x)+e^{x}f(x)

⇒∫e^{x}f(x)dx=\;\;{1}/{2}[ϕ(x)+e^{x}f(x)]

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