What is the integral of ((x^4+x))/(sqrt(x)) ?

The integral of ((x^4+x))/(sqrt(x)) is 2/9 x^{9/2}+2/3 x^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of ((x^4+x))/(sqrt(x))

\int \frac{( x ^{4} + x )}{x} d x

\frac{2}{9} x^{\frac{9}{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

\int \frac{x ^{4} + x}{x} d x

Simplify

\frac{x ^{4} + x}{x} : x (x^{3} + 1)

= \int x (x^{3} + 1) d x

Expand

x (x^{3} + 1) : x^{\frac{7}{2}} + x

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

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

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

= \frac{2}{9} x^{\frac{9}{2}} + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} x^{\frac{9}{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

\int 4 cos^{3} (x) sin (x) d x

\int ln (16 x) d x

\int (2 x^{2} + 4)^{2} d x

\int sec^{2} (θ) e^{4 t a n (θ)} d θ

\int 10 y^{5} d y

What is the integral of ((x^4+x))/(sqrt(x)) ?
The integral of ((x^4+x))/(sqrt(x)) is 2/9 x^{9/2}+2/3 x^{3/2}+C