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

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

integral of sqrt(x)(x+x^{-1})

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

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

Solution steps

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

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \int x (x + \frac{1}{x}) d x

Expand

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

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

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

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

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

Add a constant to the solution

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

\frac{d}{d x} (8 cos (x) sin (x))

y^{^{'}} = y (3 - y)

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

\frac{\partial}{\partial x} (7 yz ln (x))

t an g e n t y = \frac{3}{x}, (1, 3)

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