What is the integral of ((x+2))/((2-x)^{1/3)} ?

The integral of ((x+2))/((2-x)^{1/3)} is 3/5 (2-x)^{5/3}-6(2-x)^{2/3}+C

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integral of ((x+2))/((2-x)^{1/3)}

\int \frac{( x + 2 )}{( 2 - x ) ^{\frac{1}{3}}} d x

\frac{3}{5} (2 - x)^{\frac{5}{3}} - 6 (2 - x)^{\frac{2}{3}} + C

Solution steps

\int \frac{x + 2}{( 2 - x ) ^{\frac{1}{3}}} d x

Apply u-substitution

: \int - \frac{- u + 4}{u ^{\frac{1}{3}}} d u

= \int - \frac{- u + 4}{u ^{\frac{1}{3}}} d u

Take the constant out:

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

= - \int \frac{- u + 4}{u ^{\frac{1}{3}}} d u

Expand

\frac{- u + 4}{u ^{\frac{1}{3}}} : - u^{\frac{2}{3}} + \frac{4}{u ^{\frac{1}{3}}}

= - \int - u^{\frac{2}{3}} + \frac{4}{u ^{\frac{1}{3}}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- \int u^{\frac{2}{3}} d u + \int \frac{4}{u ^{\frac{1}{3}}} d u)

\int u^{\frac{2}{3}} d u = \frac{3}{5} u^{\frac{5}{3}}

\int \frac{4}{u ^{\frac{1}{3}}} d u = 6 u^{\frac{2}{3}}

= - (- \frac{3}{5} u^{\frac{5}{3}} + 6 u^{\frac{2}{3}})

Substitute back

u = 2 - x

= - (- \frac{3}{5} (2 - x)^{\frac{5}{3}} + 6 (2 - x)^{\frac{2}{3}})

Simplify

= \frac{3}{5} (2 - x)^{\frac{5}{3}} - 6 (2 - x)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} (2 - x)^{\frac{5}{3}} - 6 (2 - x)^{\frac{2}{3}} + C

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What is the integral of ((x+2))/((2-x)^{1/3)} ?
The integral of ((x+2))/((2-x)^{1/3)} is 3/5 (2-x)^{5/3}-6(2-x)^{2/3}+C