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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

\int \frac{1}{x} d x = 2 x

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

Add a constant to the solution

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

\frac{d}{d x} (se x^{2} (x))

\int t^{2} cos (t) d t

d er i v a t i v e (x - 9) (x - 3)^{2}

d er i v a t i v e y = x^{5 s i n (x)}

\int \frac{x \cdot arcsin ( x )}{1 - x ^{2}} d x

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