What is $f (x) = \int \frac{1}{x}$ $f(x) = int 1/x$ if $f (2) = 1$ $f(2)=1$ ?

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What is $f (x) = \int \frac{1}{x}$ $f(x) = int 1/x$ if $f (2) = 1$ $f(2)=1$ ?

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Answer 1 · 2016-05-03T19:32:16

$ln (\frac{x}{2}) + 1$ $ln(x/2)+1$

Explanation:

The derivative of $ln x = \frac{1}{x}$ $lnx=1/x$

hence the anti-derivative of $\frac{1}{x} is ln x$ $1/x" is"lnx$

$\Rightarrow F (x) = \int \frac{1}{x} d x = ln x + c$ $rArrF(x)= int1/x dx=lnx+c$

To find c , use f(2) = 1

ln2 + c = 1 → c = 1 - ln2

$\Rightarrow F (x) = ln x + 1 - ln 2$ $rArr F(x)=lnx+1-ln2$

using $∙ ln x - ln y = ln (\frac{x}{y}) to simplify$ $• lnx-lny=ln(x/y)"to simplify"$

$\Rightarrow \int \frac{1}{x} d x = ln (\frac{x}{2}) + 1$ $rArr int1/x dx= ln(x/2)+1$

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What is $f (x) = \int \frac{1}{x}$ $f(x) = int 1/x$ if $f (2) = 1$ $f(2)=1$ ?

1 Answer

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