What is the integral of sqrt(x)(sqrt(x)+1)^2 ?

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

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integral of sqrt(x)(sqrt(x)+1)^2

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

\frac{d y}{d x} = x^{3} (x - x y)

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

\int 5^{s i n (x)} cos (x) d x

n = 1 \sum \infty \frac{2 ^{n}}{e ^{n}}

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

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