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

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

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integral of (x-sqrt(x)+1)(sqrt(x)+1)

\int (x - x + 1) (x + 1) d x

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

Solution steps

\int (x - x + 1) (x + 1) d x

Expand

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

12 x^{3} - 3 y^{2} x^{4} + 1 \frac{d y}{d x} = 0

\frac{\partial}{\partial x} (cos^{5} (x^{3} y^{8}))

\frac{d ^{2} y}{d x ^{2}} = - y

f (x) = x^{2} (3 - x)^{2}

x \to - 1 lim (∣ x ∣ - 3)

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