What is the integral of a^{4x} ?

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

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integral of a^{4x}

\int a^{4 x} d x

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

Solution steps

\int a^{4 x} d x

Apply u-substitution

= \int a^{u} \frac{1}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int a^{u} d u

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

= \frac{1}{4} \cdot \frac{a ^{u}}{ln ( a )}

Substitute back

u = 4 x

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (sin (3 x^{2} - 1))

x \to 0 lim ((1 + x)^{\frac{1}{3 x}})

y = 3 x + 3 x

\frac{d y}{d x} + y = e^{x} y^{- 2}

y = x^{3}, at x = - 4