Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2/Proof 2
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Example of Use of Primitive of $\dfrac {p x + q} {a x^2 + 2 b x + c}$
- $\ds \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x = \dfrac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C$
Proof
| \(\ds \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x\) | \(=\) | \(\ds \dfrac 1 3 \int \dfrac {6 x + 3} {3 x^2 + 4 x + 2} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \int \dfrac {6 x + 4 - 1} {3 x^2 + 4 x + 2} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \paren {\int \dfrac {6 x + 4} {3 x^2 + 4 x + 2} \rd x - \int \dfrac {\d x} {3 x^2 + 4 x + 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \paren {\ln \size {3 x^2 + 4 x + 2} - \int \dfrac {\d x} {3 x^2 + 4 x + 2} }\) | Primitive of Function under its Derivative: $\map {\dfrac \d {\d x} } {3 x^2 + 4 x + 2} = 6 x + 4$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 3 \paren {\dfrac 1 {\sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } }\) | Primitive of $\dfrac 1 {3 x^2 + 4 x + 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C\) | simplifying |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Algebraic Integration: Type $\text B$.