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

The integral of (x(x+1)^2)/(sqrt(x)) is 2/7 x^{7/2}+4/5 x^{5/2}+2/3 x^{3/2}+C

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

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

\frac{2}{7} x^{\frac{7}{2}} + \frac{4}{5} x^{\frac{5}{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

Expand

\frac{x ( x + 1 ) ^{2}}{x} : x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + x

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

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

= \frac{2}{7} x^{\frac{7}{2}} + \frac{4}{5} x^{\frac{5}{2}} + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} + \frac{4}{5} x^{\frac{5}{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

y^{^{''}} - 6 y^{^{'}} + 13 = 0

(- 4, 2), (1, - 4)

\frac{d y}{d x} - 3 y = 0

y = 3 + 2 x, y = \frac{6 + x}{2}

x \to - 1 lim (x^{2} + 8 x + 8)

What is the integral of (x(x+1)^2)/(sqrt(x)) ?
The integral of (x(x+1)^2)/(sqrt(x)) is 2/7 x^{7/2}+4/5 x^{5/2}+2/3 x^{3/2}+C