integral from 2 to 5 of (2x-5)(x-3)^9

解

\int_{2}^{5} (2 x - 5) (x - 3)^{9} d x

\frac{52233}{110}

十進法表記

474.84545 \dots

解答ステップ

\int_{2}^{5} (2 x - 5) (x - 3)^{9} d x

拡張

(2 x - 5) (x - 3)^{9} : 2 x^{10} - 59 x^{9} + 783 x^{8} - 6156 x^{7} + 31752 x^{6} - 112266 x^{5} + 275562 x^{4} - 463644 x^{3} + 511758 x^{2} - 334611 x + 98415

= \int_{2}^{5} 2 x^{10} - 59 x^{9} + 783 x^{8} - 6156 x^{7} + 31752 x^{6} - 112266 x^{5} + 275562 x^{4} - 463644 x^{3} + 511758 x^{2} - 334611 x + 98415 d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{2}^{5} 2 x^{10} d x - \int_{2}^{5} 59 x^{9} d x + \int_{2}^{5} 783 x^{8} d x - \int_{2}^{5} 6156 x^{7} d x + \int_{2}^{5} 31752 x^{6} d x - \int_{2}^{5} 112266 x^{5} d x + \int_{2}^{5} 275562 x^{4} d x - \int_{2}^{5} 463644 x^{3} d x + \int_{2}^{5} 511758 x^{2} d x - \int_{2}^{5} 334611 x d x + \int_{2}^{5} 98415 d x

\int_{2}^{5} 2 x^{10} d x = \frac{97652154}{11}

\int_{2}^{5} 59 x^{9} d x = \frac{576111459}{10}

\int_{2}^{5} 783 x^{8} d x = 169877331

\int_{2}^{5} 6156 x^{7} d x = \frac{600777891}{2}

\int_{2}^{5} 31752 x^{6} d x = 353794392

\int_{2}^{5} 112266 x^{5} d x = 291161871

\int_{2}^{5} 275562 x^{4} d x = \frac{852313266}{5}

\int_{2}^{5} 463644 x^{3} d x = 70589799

\int_{2}^{5} 511758 x^{2} d x = 19958562

\int_{2}^{5} 334611 x d x = \frac{7026831}{2}

\int_{2}^{5} 98415 d x = 295245

= \frac{97652154}{11} - \frac{576111459}{10} + 169877331 - \frac{600777891}{2} + 353794392 - 291161871 + \frac{852313266}{5} - 70589799 + 19958562 - \frac{7026831}{2} + 295245

簡素化

\frac{97652154}{11} - \frac{576111459}{10} + 169877331 - \frac{600777891}{2} + 353794392 - 291161871 + \frac{852313266}{5} - 70589799 + 19958562 - \frac{7026831}{2} + 295245 : \frac{52233}{110}

= \frac{52233}{110}