What is the integral from-1 to 0 of x^3(x^2-1)^{10} ?

The integral from-1 to 0 of x^3(x^2-1)^{10} is -1/264

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integral from-1 to 0 of x^3(x^2-1)^{10}

\int_{- 1}^{0} x^{3} (x^{2} - 1)^{10} d x

- \frac{1}{264}

Decimal

- 0.00378 \dots

Solution steps

\int_{- 1}^{0} x^{3} (x^{2} - 1)^{10} d x

Expand

x^{3} (x^{2} - 1)^{10} : x^{23} - 10 x^{21} + 45 x^{19} - 120 x^{17} + 210 x^{15} - 252 x^{13} + 210 x^{11} - 120 x^{9} + 45 x^{7} - 10 x^{5} + x^{3}

= \int_{- 1}^{0} x^{23} - 10 x^{21} + 45 x^{19} - 120 x^{17} + 210 x^{15} - 252 x^{13} + 210 x^{11} - 120 x^{9} + 45 x^{7} - 10 x^{5} + x^{3} 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}^{0} x^{23} d x - \int_{- 1}^{0} 10 x^{21} d x + \int_{- 1}^{0} 45 x^{19} d x - \int_{- 1}^{0} 120 x^{17} d x + \int_{- 1}^{0} 210 x^{15} d x - \int_{- 1}^{0} 252 x^{13} d x + \int_{- 1}^{0} 210 x^{11} d x - \int_{- 1}^{0} 120 x^{9} d x + \int_{- 1}^{0} 45 x^{7} d x - \int_{- 1}^{0} 10 x^{5} d x + \int_{- 1}^{0} x^{3} d x

\int_{- 1}^{0} x^{23} d x = - \frac{1}{24}

\int_{- 1}^{0} 10 x^{21} d x = - \frac{5}{11}

\int_{- 1}^{0} 45 x^{19} d x = - \frac{9}{4}

\int_{- 1}^{0} 120 x^{17} d x = - \frac{20}{3}

\int_{- 1}^{0} 210 x^{15} d x = - \frac{105}{8}

\int_{- 1}^{0} 252 x^{13} d x = - 18

\int_{- 1}^{0} 210 x^{11} d x = - \frac{35}{2}

\int_{- 1}^{0} 120 x^{9} d x = - 12

\int_{- 1}^{0} 45 x^{7} d x = - \frac{45}{8}

\int_{- 1}^{0} 10 x^{5} d x = - \frac{5}{3}

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

= - \frac{1}{24} - (- \frac{5}{11}) - \frac{9}{4} - (- \frac{20}{3}) - \frac{105}{8} - (- 18) - \frac{35}{2} - (- 12) - \frac{45}{8} - (- \frac{5}{3}) - \frac{1}{4}

Simplify

- \frac{1}{24} - (- \frac{5}{11}) - \frac{9}{4} - (- \frac{20}{3}) - \frac{105}{8} - (- 18) - \frac{35}{2} - (- 12) - \frac{45}{8} - (- \frac{5}{3}) - \frac{1}{4} : - \frac{1}{264}

= - \frac{1}{264}

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

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\int_{0}^{2 x} e^{- x^{2}} d x

What is the integral from-1 to 0 of x^3(x^2-1)^{10} ?
The integral from-1 to 0 of x^3(x^2-1)^{10} is -1/264