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

The integral of (x^3+sqrt(x^3)+2)/(sqrt(x)) is 2/7 x^{7/2}+(x^2)/2+4sqrt(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3+sqrt(x^3)+2)/(sqrt(x))

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

\frac{2}{7} x^{\frac{7}{2}} + \frac{x ^{2}}{2} + 4 x + C

Solution steps

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

Expand

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

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

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

\int \frac{2}{x} d x = 4 x

= \frac{2}{7} x^{\frac{7}{2}} + \frac{x ^{2}}{2} + 4 x

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} + \frac{x ^{2}}{2} + 4 x + C

\int 8 x - 8 d x

\int 3 x 2 x^{2} - 4 d x

\int 6 x + 10 d x

\int 8 cos^{3} (4 x) d x

\int \frac{( x - 1 ) ( x + 5 )}{x + 2} d x

What is the integral of (x^3+sqrt(x^3)+2)/(sqrt(x)) ?
The integral of (x^3+sqrt(x^3)+2)/(sqrt(x)) is 2/7 x^{7/2}+(x^2)/2+4sqrt(x)+C