integral from 3 to 5 of (x-3)^4(5-x)^2

解答

\int_{3}^{5} (x - 3)^{4} (5 - x)^{2} d x

\frac{128}{105}

十进制

1.21904 \dots

求解步骤

\int_{3}^{5} (x - 3)^{4} (5 - x)^{2} d x

乘开

(x - 3)^{4} (5 - x)^{2} : x^{6} - 22 x^{5} + 199 x^{4} - 948 x^{3} + 2511 x^{2} - 3510 x + 2025

= \int_{3}^{5} x^{6} - 22 x^{5} + 199 x^{4} - 948 x^{3} + 2511 x^{2} - 3510 x + 2025 d x

使用积分加法定则:

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

= \int_{3}^{5} x^{6} d x - \int_{3}^{5} 22 x^{5} d x + \int_{3}^{5} 199 x^{4} d x - \int_{3}^{5} 948 x^{3} d x + \int_{3}^{5} 2511 x^{2} d x - \int_{3}^{5} 3510 x d x + \int_{3}^{5} 2025 d x

\int_{3}^{5} x^{6} d x = \frac{75938}{7}

\int_{3}^{5} 22 x^{5} d x = \frac{163856}{3}

\int_{3}^{5} 199 x^{4} d x = \frac{573518}{5}

\int_{3}^{5} 948 x^{3} d x = 128928

\int_{3}^{5} 2511 x^{2} d x = 82026

\int_{3}^{5} 3510 x d x = 28080

\int_{3}^{5} 2025 d x = 4050

= \frac{75938}{7} - \frac{163856}{3} + \frac{573518}{5} - 128928 + 82026 - 28080 + 4050

化简

\frac{75938}{7} - \frac{163856}{3} + \frac{573518}{5} - 128928 + 82026 - 28080 + 4050 : \frac{128}{105}

= \frac{128}{105}

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