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The first derivative of a function \( f(x) \) is given below.\(\frac{e^x(x -1)}{x^2}\)If f(1) = e + 3, find f(x). Show your steps. |
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Answer» \(f^1(x) = \frac{e^x(x -1)}{x^2}\) \(f(x) = \int\frac{xe^x - e^x}{x^2}dx +c\) \(= \int e^x(\frac1x - \frac1{x^2})dx +c\) \(=\frac{e^x}{x}+C\) \(\left(\therefore \int e^x (f(x) + f^1(x))dx = e^xf(x) + c \right)\) \(\left(Here\,f(x) = \frac1x⇒f^1(x) = \frac{-1}{x^2}\right)\) \(\because f (1) = e +3\) ⇒ \(e + 3 = e + c\) ⇒ \(c =3\) \(\therefore f(x) = \frac{e^x}{x} +3\) |
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