Primitive of x squared over a x + b/Examples/x^2 over x - 1

Example of Use of Primitive of $\dfrac {x^2} {a x + b}$

$\ds \int \dfrac {x^2 \rd x} {x - 1} = \frac {x^2} 2 + x + \ln \size {x - 1} + C$

Proof

From Primitive of $\dfrac {x^2} {a x + b}$:

$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$

Hence:

\(\ds \int \dfrac {x^2 \rd x} {x - 1}\)

\(=\)

\(\ds \frac {\paren {x - 1}^2} 2 + 2 \paren {x - 1} + \ln \size {x - 1} + C\)

setting $a \gets 1$, $b \gets -1$ and simplifying

\(\ds \)

\(=\)

\(\ds \frac {x^2 - 2 x + 1} 2 + 2 x - 2 + \ln \size {x - 1} + C\)

multiplying out

\(\ds \)

\(=\)

\(\ds \frac {x^2} 2 + \dfrac 1 2 + x - 2 + \ln \size {x - 1} + C\)

simplifying

\(\ds \)

\(=\)

\(\ds \frac {x^2} 2 + x + \ln \size {x - 1} + C\)

subsuming $-2 + \dfrac 1 2$ into the arbitrary constant

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $21$.