Primitive of Function under its Derivative/Examples/2 x - 5 over (x - 2) (x - 3)

Example of Use of Primitive of Function under its Derivative

$\ds \int \dfrac {2 x - 5} {\paren {x - 2} \paren {x - 3} } = \ln \size {\paren {x - 2} \paren {x - 3} } + C$

Proof

\(\ds \paren {x - 2} \paren {x - 3}\)

\(=\)

\(\ds x^2 - 5 x + 6\)

\(\ds \leadsto \ \ \)

\(\ds \map {\dfrac \d {\d x} } {\paren {x - 2} \paren {x - 3} }\)

\(=\)

\(\ds 2 x - 5\)

Power Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \int \dfrac {2 x - 5} {\paren {x - 2} \paren {x - 3} } \rd x\)

\(=\)

\(\ds \ln \size {\paren {x - 2} \paren {x - 3} } + C\)

Primitive of Function under its Derivative

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $22$.