What is the integral from 0 to k of 8^{x/k} ?

The integral from 0 to k of 8^{x/k} is (7k)/(3ln(2))

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integral from 0 to k of 8^{x/k}

\int_{0}^{k} 8^{\frac{x}{k}} d x

\frac{7 k}{3 ln ( 2 )}

Solution steps

\int_{0}^{k} 8^{\frac{x}{k}} d x

Apply u-substitution

: \int_{0}^{1} 8^{u} k d u

= \int_{0}^{1} 8^{u} k d u

Take the constant out:

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

= k \cdot \int_{0}^{1} 8^{u} d u

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

= k [\frac{8 ^{u}}{ln ( 8 )}]_{0}^{1}

Simplify

ln (8) : 3 ln (2)

= k [\frac{8 ^{u}}{3 ln ( 2 )}]_{0}^{1}

Compute the boundaries:

\frac{7}{3 ln ( 2 )}

= k \frac{7}{3 ln ( 2 )}

Simplify

= \frac{7 k}{3 ln ( 2 )}

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What is the integral from 0 to k of 8^{x/k} ?
The integral from 0 to k of 8^{x/k} is (7k)/(3ln(2))