integral of 5(x^4+5)^5

解答

\int 5 (x^{4} + 5)^{5} d x

5 (\frac{x ^{21}}{21} + \frac{25 x ^{17}}{17} + \frac{250 x ^{13}}{13} + \frac{1250 x ^{9}}{9} + 625 x^{5} + 3125 x) + C

求解步骤

\int 5 (x^{4} + 5)^{5} d x

提出常数:

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

= 5 \cdot \int (x^{4} + 5)^{5} d x

乘开

(x^{4} + 5)^{5} : x^{20} + 25 x^{16} + 250 x^{12} + 1250 x^{8} + 3125 x^{4} + 3125

= 5 \cdot \int x^{20} + 25 x^{16} + 250 x^{12} + 1250 x^{8} + 3125 x^{4} + 3125 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

= 5 (\int x^{20} d x + \int 25 x^{16} d x + \int 250 x^{12} d x + \int 1250 x^{8} d x + \int 3125 x^{4} d x + \int 3125 d x)

\int x^{20} d x = \frac{x ^{21}}{21}

\int 25 x^{16} d x = \frac{25 x ^{17}}{17}

\int 250 x^{12} d x = \frac{250 x ^{13}}{13}

\int 1250 x^{8} d x = \frac{1250 x ^{9}}{9}

\int 3125 x^{4} d x = 625 x^{5}

\int 3125 d x = 3125 x

= 5 (\frac{x ^{21}}{21} + \frac{25 x ^{17}}{17} + \frac{250 x ^{13}}{13} + \frac{1250 x ^{9}}{9} + 625 x^{5} + 3125 x)

解答补常数

= 5 (\frac{x ^{21}}{21} + \frac{25 x ^{17}}{17} + \frac{250 x ^{13}}{13} + \frac{1250 x ^{9}}{9} + 625 x^{5} + 3125 x) + C

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