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
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XV}$: $1. \ \text{(e)}$