Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2/Proof 1
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Example of Use of Primitive of $\dfrac {x^2} {a x^2 + b x + c}$
- $\ds \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x = 2 x + \dfrac 1 3 \ln \size {3 x^3 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \arctan \dfrac {3 x + 2} {\sqrt 2} + C$
Proof
| \(\ds \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x\) | \(=\) | \(\ds \int \dfrac {6 x^2 + 8 x + 4} {3 x^2 + 4 x + 2} \rd x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int 2 \rd x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x\) | Primitive of Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x + \dfrac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C\) | Primitive of $\dfrac {2 x + 1} {3 x^2 + 4 x + 2}$ |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Algebraic Integration: Type $\text C$.