What is the integral of x(x+1)^{100} ?

The integral of x(x+1)^{100} is ((x+1)^{102})/(102)-((x+1)^{101})/(101)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(x+1)^{100}

\int x (x + 1)^{100} d x

\frac{( x + 1 ) ^{102}}{102} - \frac{( x + 1 ) ^{101}}{101} + C

Solution steps

\int x (x + 1)^{100} d x

Apply u-substitution

: \int (u - 1) u^{100} d u

= \int (u - 1) u^{100} d u

Expand

(u - 1) u^{100} : u^{101} - u^{100}

= \int u^{101} - u^{100} 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^{101} d u - \int u^{100} d u

\int u^{101} d u = \frac{u ^{102}}{102}

\int u^{100} d u = \frac{u ^{101}}{101}

= \frac{u ^{102}}{102} - \frac{u ^{101}}{101}

Substitute back

u = x + 1

= \frac{( x + 1 ) ^{102}}{102} - \frac{( x + 1 ) ^{101}}{101}

Add a constant to the solution

= \frac{( x + 1 ) ^{102}}{102} - \frac{( x + 1 ) ^{101}}{101} + C

f (x) = e^{5 x}, at x = \frac{1}{5} ln (5)

x \to 4 - lim (\frac{1 - x}{4 - x})

\int x ln (9 x) d x

x \to 2 lim (x^{2} + 4)

\frac{d}{d x} (x e^{c o s (2 x - y)})

What is the integral of x(x+1)^{100} ?
The integral of x(x+1)^{100} is ((x+1)^{102})/(102)-((x+1)^{101})/(101)+C