What is the integral from 4 to 5 of 1/((x-4)^{1/3)} ?

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

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integral from 4 to 5 of 1/((x-4)^{1/3)}

\int_{4}^{5} \frac{1}{( x - 4 ) ^{\frac{1}{3}}} d x

\frac{3}{2}

Decimal Notation

1.5

Solution steps

\int_{4}^{5} \frac{1}{( x - 4 ) ^{\frac{1}{3}}} d x

Apply Integration By Parts

: [\frac{x}{( x - 4 ) ^{\frac{1}{3}}} - \int - \frac{x}{3 ( x - 4 ) ^{\frac{4}{3}}} d x]_{4}^{5}

= [\frac{x}{( x - 4 ) ^{\frac{1}{3}}} - \int - \frac{x}{3 ( x - 4 ) ^{\frac{4}{3}}} d x]_{4}^{5}

\int - \frac{x}{3 ( x - 4 ) ^{\frac{4}{3}}} d x = - \frac{x - 12}{2 ( x - 4 ) ^{\frac{1}{3}}}

= [\frac{x}{( x - 4 ) ^{\frac{1}{3}}} - (- \frac{x - 12}{2 ( x - 4 ) ^{\frac{1}{3}}})]_{4}^{5}

Simplify

[\frac{x}{( x - 4 ) ^{\frac{1}{3}}} - (- \frac{x - 12}{2 ( x - 4 ) ^{\frac{1}{3}}})]_{4}^{5} : [\frac{3}{2} (x - 4)^{\frac{2}{3}}]_{4}^{5}

= [\frac{3}{2} (x - 4)^{\frac{2}{3}}]_{4}^{5}

Compute the boundaries:

\frac{3}{2}

= \frac{3}{2}

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