What is the integral of (xsqrt(1-x)) ?

The integral of (xsqrt(1-x)) is 2/5 (1-x)^{5/2}-2/3 (1-x)^{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 (xsqrt(1-x))

\int (x 1 - x) d x

\frac{2}{5} (1 - x)^{\frac{5}{2}} - \frac{2}{3} (1 - x)^{\frac{3}{2}} + C

Solution steps

\int x 1 - x d x

Apply u-substitution

: \int - (- u + 1) u d u

= \int - (- u + 1) u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int (- u + 1) u d u

Expand

(- u + 1) u : - u^{\frac{3}{2}} + u

= - \int - u^{\frac{3}{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 u d u)

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

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

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

Substitute back

u = 1 - x

= - (- \frac{2}{5} (1 - x)^{\frac{5}{2}} + \frac{2}{3} (1 - x)^{\frac{3}{2}})

Simplify

= \frac{2}{5} (1 - x)^{\frac{5}{2}} - \frac{2}{3} (1 - x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (1 - x)^{\frac{5}{2}} - \frac{2}{3} (1 - x)^{\frac{3}{2}} + C

\int \frac{5}{x x ^{2} - 1} d x

\int (x + 12) 24 x + x^{2} d x

\int \frac{x}{x ^{4} + 3 x ^{2} + 2} d x

\int \frac{( y + 3 )}{( 3 - y ) ^{\frac{2}{3}}} d y

\int (x^{3} + 4 x^{2} - x + 5) cos (3 x) d x

What is the integral of (xsqrt(1-x)) ?
The integral of (xsqrt(1-x)) is 2/5 (1-x)^{5/2}-2/3 (1-x)^{3/2}+C