What is the integral from 0 to 8 of sqrt(1+x^{-2/3)} ?

The integral from 0 to 8 of sqrt(1+x^{-2/3)} is 5sqrt(5)-1

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integral from 0 to 8 of sqrt(1+x^{-2/3)}

\int_{0}^{8} 1 + x^{- \frac{2}{3}} d x

55 - 1

Decimal

10.18033 \dots

Solution steps

\int_{0}^{8} 1 + x^{- \frac{2}{3}} d x

Simplify

1 + x^{- \frac{2}{3}} : \frac{x ^{\frac{2}{3}} + 1}{x ^{\frac{2}{3}}}

= \int_{0}^{8} \frac{x ^{\frac{2}{3}} + 1}{x ^{\frac{2}{3}}} d x

Apply u-substitution

= \int_{1}^{5} \frac{3 u ^{\frac{1}{2}}}{2} d u

Take the constant out:

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

= \frac{3}{2} \cdot \int_{1}^{5} u^{\frac{1}{2}} d u

Apply the Power Rule

= \frac{3}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{5}

Compute the boundaries:

\frac{10 5}{3} - \frac{2}{3}

= \frac{3}{2} (\frac{10 5}{3} - \frac{2}{3})

Simplify

= 55 - 1

What is the integral from 0 to 8 of sqrt(1+x^{-2/3)} ?
The integral from 0 to 8 of sqrt(1+x^{-2/3)} is 5sqrt(5)-1