What is the integral from 1 to 2 of x(x^2-1)^3 ?

The integral from 1 to 2 of x(x^2-1)^3 is 81/8

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integral from 1 to 2 of x(x^2-1)^3

\int_{1}^{2} x (x^{2} - 1)^{3} d x

\frac{81}{8}

Decimal

10.125

Solution steps

\int_{1}^{2} x (x^{2} - 1)^{3} d x

Expand

x (x^{2} - 1)^{3} : x^{7} - 3 x^{5} + 3 x^{3} - x

= \int_{1}^{2} x^{7} - 3 x^{5} + 3 x^{3} - 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^{7} d x - \int_{1}^{2} 3 x^{5} d x + \int_{1}^{2} 3 x^{3} d x - \int_{1}^{2} x d x

\int_{1}^{2} x^{7} d x = \frac{255}{8}

\int_{1}^{2} 3 x^{5} d x = \frac{63}{2}

\int_{1}^{2} 3 x^{3} d x = \frac{45}{4}

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

= \frac{255}{8} - \frac{63}{2} + \frac{45}{4} - \frac{3}{2}

Simplify

\frac{255}{8} - \frac{63}{2} + \frac{45}{4} - \frac{3}{2} : \frac{81}{8}

= \frac{81}{8}

What is the integral from 1 to 2 of x(x^2-1)^3 ?
The integral from 1 to 2 of x(x^2-1)^3 is 81/8