What is the integral from 0 to 4 of 2^{x/2} ?

The integral from 0 to 4 of 2^{x/2} is 6/(ln(2))

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integral from 0 to 4 of 2^{x/2}

\int_{0}^{4} 2^{\frac{x}{2}} d x

\frac{6}{ln ( 2 )}

Decimal

8.65617 \dots

Solution steps

\int_{0}^{4} 2^{\frac{x}{2}} d x

Apply u-substitution

= \int_{0}^{2} 2^{u + 1} d u

2^{u + 1} = 2^{1} \cdot 2^{u}

= \int_{0}^{2} 2^{1} \cdot 2^{u} d u

Take the constant out:

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

= 2^{1} \cdot \int_{0}^{2} 2^{u} d u

Simplify

= 2 \cdot \int_{0}^{2} 2^{u} d u

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

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

Compute the boundaries:

\frac{3}{ln ( 2 )}

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

Simplify

= \frac{6}{ln ( 2 )}

What is the integral from 0 to 4 of 2^{x/2} ?
The integral from 0 to 4 of 2^{x/2} is 6/(ln(2))