What is the integral from 0 to 2 of a^{x/2} ?

The integral from 0 to 2 of a^{x/2} is (2(a-1))/(ln(a))

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integral from 0 to 2 of a^{x/2}

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

\frac{2 ( a - 1 )}{ln ( a )}

Solution steps

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

Apply u-substitution

= \int_{0}^{1} a^{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_{0}^{1} a^{u} d u

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

= 2 [\frac{a ^{u}}{ln ( a )}]_{0}^{1}

Compute the boundaries:

\frac{a}{ln ( a )} - \frac{1}{ln ( a )}

= 2 (\frac{a}{ln ( a )} - \frac{1}{ln ( a )})

Simplify

= \frac{2 ( a - 1 )}{ln ( a )}

What is the integral from 0 to 2 of a^{x/2} ?
The integral from 0 to 2 of a^{x/2} is (2(a-1))/(ln(a))