What is the integral of sqrt(x)(9x^3-2sqrt(x)+8/x) ?

The integral of sqrt(x)(9x^3-2sqrt(x)+8/x) is 16sqrt(x)+2x^{9/2}-x^2+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of sqrt(x)(9x^3-2sqrt(x)+8/x)

\int x (9 x^{3} - 2 x + \frac{8}{x}) d x

16 x + 2 x^{\frac{9}{2}} - x^{2} + C

Solution steps

\int x (9 x^{3} - 2 x + \frac{8}{x}) d x

Expand

x (9 x^{3} - 2 x + \frac{8}{x}) : \frac{8}{x} + 9 x^{\frac{7}{2}} - 2 x

= \int \frac{8}{x} + 9 x^{\frac{7}{2}} - 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 \frac{8}{x} d x + \int 9 x^{\frac{7}{2}} d x - \int 2 x d x

\int \frac{8}{x} d x = 16 x

\int 9 x^{\frac{7}{2}} d x = 2 x^{\frac{9}{2}}

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

= 16 x + 2 x^{\frac{9}{2}} - x^{2}

Add a constant to the solution

= 16 x + 2 x^{\frac{9}{2}} - x^{2} + C

What is the integral of sqrt(x)(9x^3-2sqrt(x)+8/x) ?
The integral of sqrt(x)(9x^3-2sqrt(x)+8/x) is 16sqrt(x)+2x^{9/2}-x^2+C