Question:
Integrate the function: \(e^x(\frac{1}{x}-\frac{1}{x^2})\)
Integrate the function: \(e^x(\frac{1}{x}-\frac{1}{x^2})\)
Updated On: Oct 4, 2023
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Solution and Explanation
The correct answer is: \(I=\frac{e^x}{x}+C\)
Let \(I=∫e^x[\frac{1}{x}-\frac{1}{x^2}]dx\)
Also,let \(\frac{1}{x}=ƒ(x)\,ƒ'(x)=\frac{-1}{x^2}\)
It is known that,\(∫e^x[ƒ(x)+ƒ'(x)]dx=e^x ƒ(x)+C\)
\(∴I=\frac{e^x}{x}+C\)
Let \(I=∫e^x[\frac{1}{x}-\frac{1}{x^2}]dx\)
Also,let \(\frac{1}{x}=ƒ(x)\,ƒ'(x)=\frac{-1}{x^2}\)
It is known that,\(∫e^x[ƒ(x)+ƒ'(x)]dx=e^x ƒ(x)+C\)
\(∴I=\frac{e^x}{x}+C\)
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