What is the integral of sqrt((x+1)^3) ?

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

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integral of sqrt((x+1)^3)

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

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

Solution steps

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

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

assuming

(x + 1) \geq 0

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

Apply u-substitution

: \int u^{\frac{3}{2}} d u

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = x + 1

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

Add a constant to the solution

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

\frac{d y}{d t} = \frac{( y + 2 )}{2 t + 1}

\frac{\partial}{\partial v} (3 cos (u))

- 16 (x^{- 6} - x^{12})

\frac{d}{d x} (\frac{x e ^{x}}{x ^{2} + 1})

\frac{d v}{d t} = \frac{( - 5 v - 29.4 )}{3}

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