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Popular >

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

Solution

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

Solution

\frac{1}{30}

+1

Decimal Notation

0.03333 \dots

Solution steps

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

Expand

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

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

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

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

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

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

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

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

Simplify

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

= \frac{1}{30}

Popular Examples

integral from 0 to pi of 1sin(x)

\int_{0}^{π} 1 sin (x) d x

integral from-pi/2 to pi/2 of 5cos(t)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 5 cos (t) d t

integral from 0 to x of 3/((1+x)^4)

\int_{0}^{x} \frac{3}{( 1 + x ) ^{4}} d x

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

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

integral from 0 to 3 of 1/(x^2-8x+7)

\int_{0}^{3} \frac{1}{x ^{2} - 8 x + 7} d x

Frequently Asked Questions (FAQ)

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