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Popular >

integral from 0 to 1 of x(1-x)^4

Solution

\int_{0}^{1} x (1 - x)^{4} d x

Solution

\frac{1}{30}

+1

Decimal Notation

0.03333 \dots

Solution steps

\int_{0}^{1} x (1 - x)^{4} d x

Expand

x (1 - x)^{4} : x - 4 x^{2} + 6 x^{3} - 4 x^{4} + x^{5}

= \int_{0}^{1} x - 4 x^{2} + 6 x^{3} - 4 x^{4} + x^{5} 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} 4 x^{2} d x + \int_{0}^{1} 6 x^{3} d x - \int_{0}^{1} 4 x^{4} d x + \int_{0}^{1} x^{5} d x

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

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

\int_{0}^{1} 6 x^{3} d x = \frac{3}{2}

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

\int_{0}^{1} x^{5} d x = \frac{1}{6}

= \frac{1}{2} - \frac{4}{3} + \frac{3}{2} - \frac{4}{5} + \frac{1}{6}

Simplify

\frac{1}{2} - \frac{4}{3} + \frac{3}{2} - \frac{4}{5} + \frac{1}{6} : \frac{1}{30}

= \frac{1}{30}

Popular Examples

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\int_{y}^{2} x^{3} sin (x) d x

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\int_{- 3}^{3} 2 9 - x^{2} d x

integral from-1 to 1 of 1/(x^2+x+1)

\int_{- 1}^{1} \frac{1}{x ^{2} + x + 1} d x

integral from-pi/2 to pi/2 of 7cos(x)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 7 cos (x) d x

integral from 1 to 2 of x^3+3x^2-4x

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

Frequently Asked Questions (FAQ)

What is the integral from 0 to 1 of x(1-x)^4 ?
The integral from 0 to 1 of x(1-x)^4 is 1/30