What is the integral of (a^{sqrt(t)})/(sqrt(t)) ?

The integral of (a^{sqrt(t)})/(sqrt(t)) is (2a^{sqrt(t)})/(ln(a))+C

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integral of (a^{sqrt(t)})/(sqrt(t))

\int \frac{a ^{t}}{t} d t

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

Solution steps

\int \frac{a ^{t}}{t} d t

Apply u-substitution

: \int 2 a^{u} d u

= \int 2 a^{u} d u

Take the constant out:

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

= 2 \cdot \int a^{u} d u

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

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

Substitute back

u = t

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 a ^{t}}{ln ( a )}

Add a constant to the solution

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

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