Primitive of Reciprocal of Root of a x squared plus b x plus c/Examples/2 x + x^2
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Example of Use of Primitive of $\dfrac 1 {a x^2 + b x + c}$
- $\ds \int \dfrac {\d x} {2 x + x^2} = $
Proof
We aim to use Primitive of $\dfrac 1 {a x^2 + b x + c}$ with:
\(\ds a\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds b\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 0\) |
We note that:
\(\ds b^2 - 4 a c\) | \(=\) | \(\ds 2^2 - 4 \times 1 \times 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4\) |
Hence from Primitive of $\dfrac 1 {a x^2 + b x + c}$:
- $\ds \int \frac {\d x} {a x^2 + b x + c} = \dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C$
Substituting for $a$, $b$ and $c$ and simplifying:
\(\ds \int \frac {\d x} {2 x + x^2}\) | \(=\) | \(\ds \dfrac 1 {\sqrt 1} \ln \size {2 \sqrt 1 \sqrt {1 \times x^2 + 2 \times x + 0} + 2 \times 1 \times x + 2} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {2 \paren {x + 1 + \sqrt {x^2 + 2 x} } } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln 2 + \map \ln {x + 1 + \sqrt {x^2 + 2 x} } + C\) | Sum of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {x + 1 + \sqrt {x^2 + 2 x} } + C\) | subsuming $\ln 2$ into the arbitrary constant |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $20$.