What is the integral of x(1+x)^{10} ?

The integral of x(1+x)^{10} is ((1+x)^{12})/(12)-((1+x)^{11})/(11)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(1+x)^{10}

\int x (1 + x)^{10} d x

\frac{( 1 + x ) ^{12}}{12} - \frac{( 1 + x ) ^{11}}{11} + C

Solution steps

\int x (1 + x)^{10} d x

Apply u-substitution

: \int (u - 1) u^{10} d u

= \int (u - 1) u^{10} d u

Expand

(u - 1) u^{10} : u^{11} - u^{10}

= \int u^{11} - u^{10} 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^{11} d u - \int u^{10} d u

\int u^{11} d u = \frac{u ^{12}}{12}

\int u^{10} d u = \frac{u ^{11}}{11}

= \frac{u ^{12}}{12} - \frac{u ^{11}}{11}

Substitute back

u = 1 + x

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

Add a constant to the solution

= \frac{( 1 + x ) ^{12}}{12} - \frac{( 1 + x ) ^{11}}{11} + C

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

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

\int x \cdot sin (n \cdot x) d x

\int (\frac{- 2}{x}) d x

What is the integral of x(1+x)^{10} ?
The integral of x(1+x)^{10} is ((1+x)^{12})/(12)-((1+x)^{11})/(11)+C