What is the integral from 1 to 2 of x(x-1)^4 ?

The integral from 1 to 2 of x(x-1)^4 is 11/30

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

\int_{1}^{2} x (x - 1)^{4} d x

\frac{11}{30}

Decimal

0.36666 \dots

Solution steps

\int_{1}^{2} x (x - 1)^{4} d x

Expand

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

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

\int_{1}^{2} x^{5} d x = \frac{21}{2}

\int_{1}^{2} 4 x^{4} d x = \frac{124}{5}

\int_{1}^{2} 6 x^{3} d x = \frac{45}{2}

\int_{1}^{2} 4 x^{2} d x = \frac{28}{3}

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

= \frac{21}{2} - \frac{124}{5} + \frac{45}{2} - \frac{28}{3} + \frac{3}{2}

Simplify

\frac{21}{2} - \frac{124}{5} + \frac{45}{2} - \frac{28}{3} + \frac{3}{2} : \frac{11}{30}

= \frac{11}{30}

\int ln (t) \cdot t^{2} d t

x \to + (- \infty) + lim (x^{5})

\frac{d y}{d x} = \frac{5 x}{2 y}

y = x^{- 3}, 1 \leq x \leq 6

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

What is the integral from 1 to 2 of x(x-1)^4 ?
The integral from 1 to 2 of x(x-1)^4 is 11/30