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प्रश्न
Find:
`int ((x - 3))/(x - 1)^3 e^x dx`
बेरीज
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उत्तर
`int ((x - 3))/(x - 1)^3 e^x dx`
= `int (x - 1 - 2)/(x - 1)^3 e^x dx`
= `inte^x((x - 1)/(x - 1)^3 -2/(x - 1)^3) dx`
= `int e^x (1/(x - 1)^2 - 2/(x - 1)^3) dx`
It is the form of ∫ ex [f(x) + f'(x)] dx = ex f(x) + C
Where f(x) = `1/(x - 1)^2`
∴ f'(x) = `-2/(x - 1)^3`
So, our equation becomes
`int e^x (1/(x - 1)^2 - 2/(x - 1)^3) dx = e^x/(x - 1)^2 + C`
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