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

The integral of (1-4x)^{2/3} is 3/5 x(1-4x)^{2/3}-3/20 (-4x+1)^{2/3}+C

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integral of (1-4x)^{2/3}

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

\frac{3}{5} x (1 - 4 x)^{\frac{2}{3}} - \frac{3}{20} (- 4 x + 1)^{\frac{2}{3}} + C

Solution steps

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

Apply Integration By Parts

: x (1 - 4 x)^{\frac{2}{3}} - \int - \frac{8 x}{3 ( 1 - 4 x ) ^{\frac{1}{3}}} d x

= x (1 - 4 x)^{\frac{2}{3}} - \int - \frac{8 x}{3 ( 1 - 4 x ) ^{\frac{1}{3}}} d x

\int - \frac{8 x}{3 ( 1 - 4 x ) ^{\frac{1}{3}}} d x = - \frac{1}{20} (- 8 x - 3) (- 4 x + 1)^{\frac{2}{3}}

= x (1 - 4 x)^{\frac{2}{3}} - (- \frac{1}{20} (- 8 x - 3) (- 4 x + 1)^{\frac{2}{3}})

Simplify

x (1 - 4 x)^{\frac{2}{3}} - (- \frac{1}{20} (- 8 x - 3) (- 4 x + 1)^{\frac{2}{3}}) : \frac{3}{5} x (1 - 4 x)^{\frac{2}{3}} - \frac{3}{20} (- 4 x + 1)^{\frac{2}{3}}

= \frac{3}{5} x (1 - 4 x)^{\frac{2}{3}} - \frac{3}{20} (- 4 x + 1)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} x (1 - 4 x)^{\frac{2}{3}} - \frac{3}{20} (- 4 x + 1)^{\frac{2}{3}} + C

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What is the integral of (1-4x)^{2/3} ?
The integral of (1-4x)^{2/3} is 3/5 x(1-4x)^{2/3}-3/20 (-4x+1)^{2/3}+C