Constant of integration

The problem "Fundamental Rule of Leibniz?" greatly confused me about the constant of integration. In the question, $f\left( x \right) =\int { \frac { 1 }{ x } } dx\quad and$ $f\left( -1 \right) =1$ We are asked to find the value of $f\left( 1 \right)$ According to the answer, the value cannot be found. $f\left( x \right) =\ln { \left| x \right| } +C$ $f\left( -1 \right) =\ln { \left| -1 \right| } +C=1$ $This\quad gives\quad C=1$ $Now,\quad f\left( 1 \right) =\ln { \left| 1 \right| } +C'\quad =C'$ I think that C should be equal to C' but according to the answer, $C\neq C'$ I am unable to understand it. Can someone please explain it.

#Calculus

Easy Math Editor

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Comments

I think the issue may be the lack of continuity of the function $\dfrac{1}{x}$ at $x = 0.$ First, I'm wondering if the initial statement of the question should in fact be

$\displaystyle f(x) = \int_{a}^{x} \dfrac{1}{t} dt$ with $f(-1) = 1$ for some non-zero real value $a.$

Then $f(x) = \ln |x| - ln |a| \Longrightarrow f(-1) = \ln |-1| - \ln |a| = 1 \Longrightarrow \ln |a| = -1 \Longrightarrow a = \pm \dfrac{1}{e}.$

This interpretation is still consistent with $C = 1,$ but I have phrased the question so that $C = -\ln |a|.$ Further, to avoid the discontinuity of $f(x)$ at $x = 0,$ for negative values of $x$ we would need to make $a = -\dfrac{1}{e},$ and for positive values of $x$ make $a = \dfrac{1}{e}.$ This may seem like an odd requirement, but in order to make $f(x)$ well-defined for both positive and negative values of $x$ I think that this is necessary.

So with this in mind, we have that $f(-1) = \displaystyle\int_{-\frac{1}{e}}^{-1} \dfrac{1}{t} dt = \ln |-1| - \ln |-\dfrac{1}{e}| = 1,$ and

$f(1) = \displaystyle\int_{\frac{1}{e}}^{1} \dfrac{1}{t} dt = \ln(1) - \ln(\dfrac{1}{e}) = 1.$

So with this interpretation we, in essence, find that $C = C'$ , in agreement with your observation, but I had to rephrase the question so that $f(x)$ was well-defined for both positive and negative reals in the first place.

I may have confused you even further with this analysis, but feel free to ask any questions you might have. :)

Brian Charlesworth - 6 years, 1 month ago

Sir @Brian Charlesworth and Sir @Sandeep Bhardwaj please help.

Abhijeet Verma - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$