What is the integral of a^{bx} ?

The integral of a^{bx} is (a^{bx})/(bln(a))+C

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integral of a^{bx}

\int a^{b x} d x

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

Solution steps

\int a^{b x} d x

Apply u-substitution

: \int \frac{a ^{u}}{b} d u

= \int \frac{a ^{u}}{b} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{b} \cdot \int a^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{b} \cdot \frac{a ^{u}}{ln ( a )}

Substitute back

u = b x

= \frac{1}{b} \cdot \frac{a ^{b x}}{ln ( a )}

Simplify

\frac{1}{b} \cdot \frac{a ^{b x}}{ln ( a )} : \frac{a ^{b x}}{b ln ( a )}

= \frac{a ^{b x}}{b ln ( a )}

Add a constant to the solution

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

\int \frac{ln ( 2 x )}{2 x} d x

\int \frac{1}{10} x \cdot ln (x) d x

\int \frac{- x}{x ^{2} + 1} d x

sin (x) + cos (x)