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
Jump to navigation
Jump to search
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 1
| \(\ds \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x\) | \(=\) | \(\ds \int \dfrac {6 x^2 + 8 x + 4} {3 x^2 + 4 x + 2} \rd x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x\) | 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 $\dfrac {2 x + 1} {3 x^2 + 4 x + 2}$ |
$\blacksquare$
Proof 2
| \(\ds \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x\) | \(=\) | \(\ds 6 \int \dfrac {x^2} {3 x^2 + 4 x + 2} \rd x + 10 \int \dfrac x {3 x^2 + 4 x + 2} \rd x + 5 \int \dfrac {\d x} {3 x^2 + 4 x + 2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \paren {\frac x 3 - \frac 4 {2 \times 3^2} \ln \size {3 x^2 + 4 x + 2} + \frac {4^2 - 2 \times 3 \times 2} {2 \times 3^2} \int \frac {\d x} {3 x^2 + 4 x + 2} } + 10 \int \dfrac x {3 x^2 + 4 x + 2} \rd x + 5 \int \dfrac {\d x} {3 x^2 + 4 x + 2}\) | Primitive of $\dfrac {x^2} {a x^2 + b x + c}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x - \frac 4 3 \ln \size {3 x^2 + 4 x + 2} + \frac 4 3 \int \frac {\d x} {3 x^2 + 4 x + 2} + 10 \int \dfrac x {3 x^2 + 4 x + 2} \rd x + 5 \int \dfrac {\d x} {3 x^2 + 4 x + 2}\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x - \frac 4 3 \ln \size {3 x^2 + 4 x + 2} + 10 \int \dfrac x {3 x^2 + 4 x + 2} \rd x + \dfrac {19} 3 \int \dfrac {\d x} {3 x^2 + 4 x + 2}\) | further simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x - \frac 4 3 \ln \size {3 x^2 + 4 x + 2} + 10 \paren {\frac 1 {2 \times 3} \ln \size {3 x^2 + 4 x + 2} - \frac 4 {2 \times 3} \int \frac {\d x} {3 x^2 + 4 x + 2} } + \dfrac {19} 3 \int \dfrac {\d x} {3 x^2 + 4 x + 2}\) | Primitive of $\dfrac x {a x^2 + b x + c}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x - \frac 4 3 \ln \size {3 x^2 + 4 x + 2} + \frac 5 3 \ln \size {3 x^2 + 4 x + 2} - \frac {20} 3 \int \frac {\d x} {3 x^2 + 4 x + 2} + \dfrac {19} 3 \int \dfrac {\d x} {3 x^2 + 4 x + 2}\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x + \frac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 3 \int \frac {\d x} {3 x^2 + 4 x + 2}\) | further simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x + \frac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 3 \paren {\dfrac 1 {\sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } } + C\) | Primitive of $\dfrac 1 {3 x^2 + 4 x + 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 x + \frac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C\) | simplification |
$\blacksquare$