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Popular >

integral from 0 to 1 of x(1-x)^3

Solution

\int_{0}^{1} x (1 - x)^{3} d x

Solution

\frac{1}{20}

+1

Decimal Notation

0.05

Solution steps

\int_{0}^{1} x (1 - x)^{3} d x

Expand

x (1 - x)^{3} : x - 3 x^{2} + 3 x^{3} - x^{4}

= \int_{0}^{1} x - 3 x^{2} + 3 x^{3} - x^{4} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} x d x - \int_{0}^{1} 3 x^{2} d x + \int_{0}^{1} 3 x^{3} d x - \int_{0}^{1} x^{4} d x

\int_{0}^{1} x d x = \frac{1}{2}

\int_{0}^{1} 3 x^{2} d x = 1

\int_{0}^{1} 3 x^{3} d x = \frac{3}{4}

\int_{0}^{1} x^{4} d x = \frac{1}{5}

= \frac{1}{2} - 1 + \frac{3}{4} - \frac{1}{5}

Simplify

\frac{1}{2} - 1 + \frac{3}{4} - \frac{1}{5} : \frac{1}{20}

= \frac{1}{20}

Popular Examples

integral from 0 to 4 of 1/(x-2)

\int_{0}^{4} \frac{1}{x - 2} d x

integral from 3 to infinity of 1/(x^5)

\int_{3}^{\infty} \frac{1}{x ^{5}} d x

integral from 0 to 1 of e^{tx}2x

\int_{0}^{1} e^{t x} 2 x d x

integral from 0 to 2 of 8x-x^4

\int_{0}^{2} 8 x - x^{4} d x

integral from 2 to 4 of (2x+3)

\int_{2}^{4} (2 x + 3) d x

Frequently Asked Questions (FAQ)

What is the integral from 0 to 1 of x(1-x)^3 ?
The integral from 0 to 1 of x(1-x)^3 is 1/20