Primitive of x squared over a x + b/Proof 1
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Theorem
- $\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$
Proof
Put $u = a x + b$.
Then:
\(\ds x\) | \(=\) | \(\ds \frac {u - b} a\) | ||||||||||||
\(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac 1 a\) |
Then:
\(\ds \int \frac {x^2 \rd x} {a x + b}\) | \(=\) | \(\ds \int \frac 1 a \paren {\frac {u - b} a}^2 \frac {\d u} u\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^3} \int \frac {\paren {u - b}^2} u \rd u\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^3} \int \paren {u - 2 b + \frac {b^2} u} \rd u\) | multiplying out and simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^3} \paren {\int u \rd u - \int 2 b \rd u + \int \frac {b^2} u \rd u}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^3} \paren {\frac {u^2} 2 + C - \int 2 b \rd u + \int \frac {b^2} u \rd u}\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^3} \paren {\frac {u^2} 2 - 2 b u + C + \int \frac {b^2} u \rd u}\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^3} \paren {\frac {u^2} 2 - 2 b u + C + b^2 \int \frac {\d u} u}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^3} \paren {\frac {u^2} 2 - 2 b u + b^2 \ln \size u} + C\) | Primitive of Reciprocal and subsuming arbitrary constant $C$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C\) | substituting for $u$ and simplifying |
$\blacksquare$