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

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

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

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

x + 2 x^{\frac{3}{2}} + \frac{3 x ^{2}}{2} + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

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

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

Apply the Sum Rule:

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

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

\int 1 d x = x

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

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

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

= x + 2 x^{\frac{3}{2}} + \frac{3 x ^{2}}{2} + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= x + 2 x^{\frac{3}{2}} + \frac{3 x ^{2}}{2} + \frac{2}{5} x^{\frac{5}{2}} + C

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

f (x) = \frac{x cos ( x ) + π}{x - π}

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

\frac{\partial}{\partial x} ((ln (x))^{2})

\int \frac{x ^{3} + x ^{2} - x ^{5}}{x ^{6}} d x

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