What is the integral of x(5-x)^{11} ?

The integral of x(5-x)^{11} is -1/156 (-x+5)^{12}(12x+5)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(5-x)^{11}

\int x (5 - x)^{11} d x

- \frac{1}{156} (- x + 5)^{12} (12 x + 5) + C

Solution steps

\int x (5 - x)^{11} d x

Apply u-substitution

Take the constant out:

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

Expand

= - \frac{1}{11} \cdot \int - u^{\frac{2}{11}} + 5 u^{\frac{1}{11}} 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{1}{11} (- \int u^{\frac{2}{11}} d u + \int 5 u^{\frac{1}{11}} d u)

\int u^{\frac{2}{11}} d u = \frac{11}{13} u^{\frac{13}{11}}

\int 5 u^{\frac{1}{11}} d u = \frac{55}{12} u^{\frac{12}{11}}

= - \frac{1}{11} (- \frac{11}{13} u^{\frac{13}{11}} + \frac{55}{12} u^{\frac{12}{11}})

Substitute back

u = (5 - x)^{11}

= - \frac{1}{11} (- \frac{11}{13} ((5 - x)^{11})^{\frac{13}{11}} + \frac{55}{12} ((5 - x)^{11})^{\frac{12}{11}})

Simplify

- \frac{1}{11} (- \frac{11}{13} ((5 - x)^{11})^{\frac{13}{11}} + \frac{55}{12} ((5 - x)^{11})^{\frac{12}{11}}) : - \frac{1}{156} (- x + 5)^{12} (12 x + 5)

= - \frac{1}{156} (- x + 5)^{12} (12 x + 5)

Add a constant to the solution

= - \frac{1}{156} (- x + 5)^{12} (12 x + 5) + C

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

\int (8 x^{3} - 12 x^{2} + 4 x) d x

\int \frac{x + 1}{x x - 2} d x

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

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

What is the integral of x(5-x)^{11} ?
The integral of x(5-x)^{11} is -1/156 (-x+5)^{12}(12x+5)+C