Primitive of General Exponential of a x

Theorem

$\ds \int b^{a x} \rd x = \frac {b^{a x} } {a \ln b} + C$

where:

$a \ne 0$

$b > 0, b \ne 1$

Proof

\(\ds \int b^x \rd x\)

\(=\)

\(\ds \frac {b^x} {\ln b} + C\)

Primitive of $b^x$

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \frac 1 a \paren {\frac {b^x} {\ln b} } + C\)

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

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

simplifying

$\blacksquare$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $103$.