What is the integral of 3^{x/2} ?

The integral of 3^{x/2} is (2*3^{x/2})/(ln(3))+C

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integral of 3^{x/2}

\int 3^{\frac{x}{2}} d x

\frac{2 \cdot 3 ^{\frac{x}{2}}}{ln ( 3 )} + C

Solution steps

\int 3^{\frac{x}{2}} d x

Apply u-substitution

= \int 3^{u} \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int 3^{u} d u

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

= 2 \cdot \frac{3 ^{u}}{ln ( 3 )}

Substitute back

u = \frac{x}{2}

= 2 \cdot \frac{3 ^{\frac{x}{2}}}{ln ( 3 )}

Multiply fractions:

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

= \frac{2 \cdot 3 ^{\frac{x}{2}}}{ln ( 3 )}

Add a constant to the solution

= \frac{2 \cdot 3 ^{\frac{x}{2}}}{ln ( 3 )} + C

What is the integral of 3^{x/2} ?
The integral of 3^{x/2} is (2*3^{x/2})/(ln(3))+C