What is the integral of a^{ny} ?

The integral of a^{ny} is (a^{ny})/(nln(a))+C

integral of a^{ny}

\int a^{n y} d y

\frac{a ^{n y}}{n ln ( a )} + C

Solution steps

\int a^{n y} d y

Apply u-substitution

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

Take the constant out:

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

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

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

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

Substitute back

u = n y

= \frac{1}{n} \cdot \frac{a ^{n y}}{ln ( a )}

Simplify

\frac{1}{n} \cdot \frac{a ^{n y}}{ln ( a )} : \frac{a ^{n y}}{n ln ( a )}

= \frac{a ^{n y}}{n ln ( a )}

Add a constant to the solution

= \frac{a ^{n y}}{n ln ( a )} + C

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