If f ' ( x ) = 1 x + x and f ( 1 ) = 5 2 , then f ( x ) =

Question

If $f' (x) = \frac{1}{x} + x$ and $f (1) = \frac{5}{2}$ , then $f (x) =$

Smart Achievers · Accepted Answer

(1)

Ans. $\log x + \frac{x^{2}}{2} + 2$

Explanation

$f (x) = \int f' (x) d x = \int (\frac{1}{x} + x) d x$

$= \log x + \frac{x^{2}}{2} + C$

$f (1) = \log 1 + \frac{1^{2}}{2} + C$

$\Rightarrow$ $\frac{5}{2} = 0 + \frac{1}{2} + C$

$\Rightarrow$ $C = 2$

$∴$ $f (x) = \log x + \frac{x^{2}}{2} + 2$

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