What is the integral from 0 to 1 of 3x(x^2-1)^{13} ?

The integral from 0 to 1 of 3x(x^2-1)^{13} is -3/28

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integral from 0 to 1 of 3x(x^2-1)^{13}

\int_{0}^{1} 3 x (x^{2} - 1)^{13} d x

- \frac{3}{28}

Decimal

- 0.10714 \dots

Solution steps

\int_{0}^{1} 3 x (x^{2} - 1)^{13} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int_{0}^{1} x (x^{2} - 1)^{13} d x

Expand

x (x^{2} - 1)^{13} : x^{27} - 13 x^{25} + 78 x^{23} - 286 x^{21} + 715 x^{19} - 1287 x^{17} + 1716 x^{15} - 1716 x^{13} + 1287 x^{11} - 715 x^{9} + 286 x^{7} - 78 x^{5} + 13 x^{3} - x

= 3 \cdot \int_{0}^{1} x^{27} - 13 x^{25} + 78 x^{23} - 286 x^{21} + 715 x^{19} - 1287 x^{17} + 1716 x^{15} - 1716 x^{13} + 1287 x^{11} - 715 x^{9} + 286 x^{7} - 78 x^{5} + 13 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

= 3 (\int_{0}^{1} x^{27} d x - \int_{0}^{1} 13 x^{25} d x + \int_{0}^{1} 78 x^{23} d x - \int_{0}^{1} 286 x^{21} d x + \int_{0}^{1} 715 x^{19} d x - \int_{0}^{1} 1287 x^{17} d x + \int_{0}^{1} 1716 x^{15} d x - \int_{0}^{1} 1716 x^{13} d x + \int_{0}^{1} 1287 x^{11} d x - \int_{0}^{1} 715 x^{9} d x + \int_{0}^{1} 286 x^{7} d x - \int_{0}^{1} 78 x^{5} d x + \int_{0}^{1} 13 x^{3} d x - \int_{0}^{1} x d x)

\int_{0}^{1} x^{27} d x = \frac{1}{28}

\int_{0}^{1} 13 x^{25} d x = \frac{1}{2}

\int_{0}^{1} 78 x^{23} d x = \frac{13}{4}

\int_{0}^{1} 286 x^{21} d x = 13

\int_{0}^{1} 715 x^{19} d x = \frac{143}{4}

\int_{0}^{1} 1287 x^{17} d x = \frac{143}{2}

\int_{0}^{1} 1716 x^{15} d x = \frac{429}{4}

\int_{0}^{1} 1716 x^{13} d x = \frac{858}{7}

\int_{0}^{1} 1287 x^{11} d x = \frac{429}{4}

\int_{0}^{1} 715 x^{9} d x = \frac{143}{2}

\int_{0}^{1} 286 x^{7} d x = \frac{143}{4}

\int_{0}^{1} 78 x^{5} d x = 13

\int_{0}^{1} 13 x^{3} d x = \frac{13}{4}

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

= 3 (\frac{1}{28} - \frac{1}{2} + \frac{13}{4} - 13 + \frac{143}{4} - \frac{143}{2} + \frac{429}{4} - \frac{858}{7} + \frac{429}{4} - \frac{143}{2} + \frac{143}{4} - 13 + \frac{13}{4} - \frac{1}{2})

Simplify

3 (\frac{1}{28} - \frac{1}{2} + \frac{13}{4} - 13 + \frac{143}{4} - \frac{143}{2} + \frac{429}{4} - \frac{858}{7} + \frac{429}{4} - \frac{143}{2} + \frac{143}{4} - 13 + \frac{13}{4} - \frac{1}{2}) : - \frac{3}{28}

= - \frac{3}{28}

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What is the integral from 0 to 1 of 3x(x^2-1)^{13} ?
The integral from 0 to 1 of 3x(x^2-1)^{13} is -3/28