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

The integral of (2x+1)^{(-1)/3} is x/((2x+1)^{1/3)}+(2x+3)/(4(2x+1)^{1/3)}+C

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

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

\frac{x}{( 2 x + 1 ) ^{\frac{1}{3}}} + \frac{2 x + 3}{4 ( 2 x + 1 ) ^{\frac{1}{3}}} + C

Solution steps

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

Apply Integration By Parts

: \frac{x}{( 2 x + 1 ) ^{\frac{1}{3}}} - \int - \frac{2 x}{3 ( 2 x + 1 ) ^{\frac{4}{3}}} d x

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

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

= \frac{x}{( 2 x + 1 ) ^{\frac{1}{3}}} - (- \frac{2 x + 3}{4 ( 2 x + 1 ) ^{\frac{1}{3}}})

Simplify

= \frac{x}{( 2 x + 1 ) ^{\frac{1}{3}}} + \frac{2 x + 3}{4 ( 2 x + 1 ) ^{\frac{1}{3}}}

Add a constant to the solution

= \frac{x}{( 2 x + 1 ) ^{\frac{1}{3}}} + \frac{2 x + 3}{4 ( 2 x + 1 ) ^{\frac{1}{3}}} + C

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What is the integral of (2x+1)^{(-1)/3} ?
The integral of (2x+1)^{(-1)/3} is x/((2x+1)^{1/3)}+(2x+3)/(4(2x+1)^{1/3)}+C