What is the integral of sqrt((4x+2)^3) ?

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

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integral of sqrt((4x+2)^3)

\int (4 x + 2)^{3} d x

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

Solution steps

\int (4 x + 2)^{3} d x

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

assuming

(4 x + 2) \geq 0

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

Apply Integration By Parts

= (4 x + 2)^{\frac{3}{2}} x - \int 6 4 x + 2 x d x

\int 6 4 x + 2 x d x = \frac{2 2}{5} (3 x - 1) (2 x + 1)^{\frac{3}{2}}

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

Add a constant to the solution

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

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