What is the integral from 0 to 8 of 1/((10-2x)^{1/5)} ?

The integral from 0 to 8 of 1/((10-2x)^{1/5)} is (5*5^{4/5}-5*3^{4/5})/(4*2^{1/5)}

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integral from 0 to 8 of 1/((10-2x)^{1/5)}

\int_{0}^{8} \frac{1}{( 10 - 2 x ) ^{\frac{1}{5}}} d x

\frac{5 \cdot 5 ^{\frac{4}{5}} - 5 \cdot 3 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}}

Decimal Notation

1.32288 \dots

Solution steps

\int_{0}^{8} \frac{1}{( 10 - 2 x ) ^{\frac{1}{5}}} d x

Find undefined (singularity) points:

x = 5

If exist

b, a < b < c, f (b) =

undefined

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{5} \frac{1}{( 10 - 2 x ) ^{\frac{1}{5}}} d x + \int_{5}^{8} \frac{1}{( 10 - 2 x ) ^{\frac{1}{5}}} d x

\int_{0}^{5} \frac{1}{( 10 - 2 x ) ^{\frac{1}{5}}} d x = \frac{5 \cdot 5 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}}

\int_{5}^{8} \frac{1}{( 10 - 2 x ) ^{\frac{1}{5}}} d x = - \frac{5 ( - 1 ) ^{\frac{4}{5}} \cdot 3 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}}

= \frac{5 \cdot 5 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}} - \frac{5 ( - 1 ) ^{\frac{4}{5}} \cdot 3 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}}

Simplify

\frac{5 \cdot 5 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}} - \frac{5 ( - 1 ) ^{\frac{4}{5}} \cdot 3 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}} : \frac{5 \cdot 5 ^{\frac{4}{5}} - 5 \cdot 3 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}}

= \frac{5 \cdot 5 ^{\frac{4}{5}} - 5 \cdot 3 ^{\frac{4}{5}}}{4 \cdot 2 ^{\frac{1}{5}}}

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What is the integral from 0 to 8 of 1/((10-2x)^{1/5)} ?
The integral from 0 to 8 of 1/((10-2x)^{1/5)} is (5*5^{4/5}-5*3^{4/5})/(4*2^{1/5)}