Definite Integral/Examples/Reciprocal of 1 - x from 2 to 3

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Example of Definite Integral

$\ds \int_2^3 \dfrac {\d x} {1 - x} = \ln \dfrac 1 2$


Proof

\(\ds z\) \(=\) \(\ds 1 - x\) with a view to making a substitution
\(\ds \leadsto \ \ \) \(\ds \d z\) \(=\) \(\ds -\rd x\)

Then we re-evaluate the limits:

\(\ds 1 - 2\) \(=\) \(\ds -1\)
\(\ds 1 - 3\) \(=\) \(\ds -2\)


Hence:

\(\ds \int_2^3 \dfrac {\d x} {1 - x}\) \(=\) \(\ds \int_{-1}^{-2} \dfrac {-\rd z} z\) Integration by Substitution
\(\ds \) \(=\) \(\ds \int_{-2}^{-1} \dfrac {\d z} z\) Reversal of Limits of Definite Integral
\(\ds \) \(=\) \(\ds \bigintlimits {\ln \size z} {-2} {-1}\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \ln 1 - \ln 2\) Definition of Definite Integral
\(\ds \) \(=\) \(\ds 0 - \ln 2\) Natural Logarithm of 1 is 0
\(\ds \) \(=\) \(\ds \ln \dfrac 1 2\) Logarithm of Reciprocal

$\blacksquare$


Sources