What is the integral of x*sqrt(x-2) ?

The integral of x*sqrt(x-2) is 2/5 (x-2)^{5/2}+4/3 (x-2)^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x*sqrt(x-2)

\int x \cdot x - 2 d x

\frac{2}{5} (x - 2)^{\frac{5}{2}} + \frac{4}{3} (x - 2)^{\frac{3}{2}} + C

Solution steps

\int x x - 2 d x

Apply u-substitution

: \int (u + 2) u d u

= \int (u + 2) u d u

Expand

(u + 2) u : u^{\frac{3}{2}} + 2 u

= \int u^{\frac{3}{2}} + 2 u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{\frac{3}{2}} d u + \int 2 u d u

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

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}}

Substitute back

u = x - 2

= \frac{2}{5} (x - 2)^{\frac{5}{2}} + \frac{4}{3} (x - 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x - 2)^{\frac{5}{2}} + \frac{4}{3} (x - 2)^{\frac{3}{2}} + C

\int - \frac{1}{8} e^{- 2 x} d x

\int (\frac{sin ( 2 x )}{1 + cos ^{2} ( x )}) d x

\int \frac{x ^{3}}{x + 4} d x

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

\int 4 x (x + 2)^{7} d x

What is the integral of x*sqrt(x-2) ?
The integral of x*sqrt(x-2) is 2/5 (x-2)^{5/2}+4/3 (x-2)^{3/2}+C