What is the integral of (a^{2x})/(2ln(a)) ?

The integral of (a^{2x})/(2ln(a)) is (a^{2x})/(4ln^2(a))+C

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integral of (a^{2x})/(2ln(a))

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

\frac{a ^{2 x}}{4 ln ^{2} ( a )} + C

Solution steps

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

Take the constant out:

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

= \frac{1}{2 ln ( a )} \cdot \int a^{2 x} d x

Apply u-substitution

: \int a^{u} \frac{1}{2} d u

= \frac{1}{2 ln ( a )} \cdot \int a^{u} \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2 ln ( a )} \cdot \frac{1}{2} \cdot \int a^{u} d u

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

= \frac{1}{2 ln ( a )} \cdot \frac{1}{2} \cdot \frac{a ^{u}}{ln ( a )}

Substitute back

u = 2 x

= \frac{1}{2 ln ( a )} \cdot \frac{1}{2} \cdot \frac{a ^{2 x}}{ln ( a )}

Simplify

\frac{1}{2 ln ( a )} \cdot \frac{1}{2} \cdot \frac{a ^{2 x}}{ln ( a )} : \frac{a ^{2 x}}{4 ln ^{2} ( a )}

= \frac{a ^{2 x}}{4 ln ^{2} ( a )}

Add a constant to the solution

= \frac{a ^{2 x}}{4 ln ^{2} ( a )} + C

\int_{0}^{2} 2 x d x

\frac{d}{d x} (\frac{x}{8 + x})

\frac{d y}{d x} + \frac{y}{x} = 6 x^{7} y^{2}

\frac{d}{d x} (e^{x} + e^{e} + e x)

\frac{9 t}{4 + t ^{2}}

What is the integral of (a^{2x})/(2ln(a)) ?
The integral of (a^{2x})/(2ln(a)) is (a^{2x})/(4ln^2(a))+C