Primitive of Function under its Derivative/Examples/2 x + a over x^2 + a x + b

Example of Use of Primitive of Function under its Derivative

$\ds \int \dfrac {2 x + a} {x^2 + a x + b} \rd x = \ln \size {x^2 + a x + b} + C$

Proof

\(\ds \map {\dfrac \d {\d x} } {x^2 + a x + b}\)

\(=\)

\(\ds 2 x + a\)

Power Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \int \dfrac {2 x + a} {x^2 + a x + b} \rd x\)

\(=\)

\(\ds \ln \size {x^2 + a x + b} + C\)

Primitive of Function under its Derivative

$\blacksquare$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals