What is the integral of x(a-x)^{5/2} ?

The integral of x(a-x)^{5/2} is -2/63 (7x+2a)(-x+a)^{7/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(a-x)^{5/2}

\int x (a - x)^{\frac{5}{2}} d x

- \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}} + C

Solution steps

\int x (a - x)^{\frac{5}{2}} d x

Apply u-substitution

: \int - \frac{2 u ^{\frac{2}{5}} ( - u ^{\frac{2}{5}} + a )}{5} d u

= \int - \frac{2 u ^{\frac{2}{5}} ( - u ^{\frac{2}{5}} + a )}{5} d u

Take the constant out:

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

= - \frac{2}{5} \cdot \int u^{\frac{2}{5}} (- u^{\frac{2}{5}} + a) d u

Expand

u^{\frac{2}{5}} (- u^{\frac{2}{5}} + a) : - u^{\frac{4}{5}} + a u^{\frac{2}{5}}

= - \frac{2}{5} \cdot \int - u^{\frac{4}{5}} + a u^{\frac{2}{5}} 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

= - \frac{2}{5} (- \int u^{\frac{4}{5}} d u + \int a u^{\frac{2}{5}} d u)

\int u^{\frac{4}{5}} d u = \frac{5}{9} u^{\frac{9}{5}}

\int a u^{\frac{2}{5}} d u = a \frac{5}{7} u^{\frac{7}{5}}

= - \frac{2}{5} (- \frac{5}{9} u^{\frac{9}{5}} + a \frac{5}{7} u^{\frac{7}{5}})

Substitute back

u = (a - x)^{\frac{5}{2}}

= - \frac{2}{5} (- \frac{5}{9} ((a - x)^{\frac{5}{2}})^{\frac{9}{5}} + a \frac{5}{7} ((a - x)^{\frac{5}{2}})^{\frac{7}{5}})

Simplify

- \frac{2}{5} (- \frac{5}{9} ((a - x)^{\frac{5}{2}})^{\frac{9}{5}} + a \frac{5}{7} ((a - x)^{\frac{5}{2}})^{\frac{7}{5}}) : - \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}}

= - \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}}

Add a constant to the solution

= - \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}} + C

\int \frac{2}{x ^{10}} d x

\frac{d}{d x} (ln (14) \cdot 4 9^{x} \cdot 2^{2 x + 1})

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

\frac{\partial}{\partial x} (\frac{e ^{x}}{e ^{x} + e ^{y}})

\frac{\partial}{\partial y} (x y arcsin (yz))

What is the integral of x(a-x)^{5/2} ?
The integral of x(a-x)^{5/2} is -2/63 (7x+2a)(-x+a)^{7/2}+C