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

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

Popular >

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

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

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

Solution steps

\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

\int x^{2} d x

x \to 0 + lim (\frac{2 x}{∣ x ∣})

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

\frac{d}{d x} ((x + 2)^{2} \cdot \frac{x ^{2}}{2})

\frac{d}{d x} (\frac{e ^{x}}{5 x ^{3}})

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