What is the integral from 0 to 1 of (1-sqrt(x))^{50} ?

The integral from 0 to 1 of (1-sqrt(x))^{50} is 1/1326

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1 of (1-sqrt(x))^{50}

\int_{0}^{1} (1 - x)^{50} d x

\frac{1}{1326}

Decimal

0.00075 \dots

Solution steps

\int_{0}^{1} (1 - x)^{50} d x

Apply u-substitution

: \int_{1}^{0} - 2 u^{50} (- u + 1) d u

= \int_{1}^{0} - 2 u^{50} (- u + 1) d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{1} - 2 u^{50} (- u + 1) d u

Take the constant out:

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

= - (- 2 \cdot \int_{0}^{1} u^{50} (- u + 1) d u)

Expand

u^{50} (- u + 1) : - u^{51} + u^{50}

= - (- 2 \cdot \int_{0}^{1} - u^{51} + u^{50} 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

= - (- 2 (- \int_{0}^{1} u^{51} d u + \int_{0}^{1} u^{50} d u))

\int_{0}^{1} u^{51} d u = \frac{1}{52}

\int_{0}^{1} u^{50} d u = \frac{1}{51}

= - (- 2 (- \frac{1}{52} + \frac{1}{51}))

Simplify

- (- 2 (- \frac{1}{52} + \frac{1}{51})) : \frac{1}{1326}

= \frac{1}{1326}

x \to a - lim (f (x))

\int_{0}^{1} 7 \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

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

sin (x) + \frac{1}{9} cot (x)

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

What is the integral from 0 to 1 of (1-sqrt(x))^{50} ?
The integral from 0 to 1 of (1-sqrt(x))^{50} is 1/1326