integral of (x^5+4)^{12}x^4

解答

\int (x^{5} + 4)^{12} x^{4} d x

\frac{x ^{65}}{65} + \frac{4 x ^{60}}{5} + \frac{96 x ^{55}}{5} + \frac{1408 x ^{50}}{5} + 2816 x^{45} + \frac{101376 x ^{40}}{5} + \frac{540672 x ^{35}}{5} + \frac{2162688 x ^{30}}{5} + \frac{6488064 x ^{25}}{5} + 2883584 x^{20} + \frac{23068672 x ^{15}}{5} + \frac{25165824 x ^{10}}{5} + \frac{16777216 x ^{5}}{5} + C

求解步骤

\int (x^{5} + 4)^{12} x^{4} d x

乘开

(x^{5} + 4)^{12} x^{4} : x^{64} + 48 x^{59} + 1056 x^{54} + 14080 x^{49} + 126720 x^{44} + 811008 x^{39} + 3784704 x^{34} + 12976128 x^{29} + 32440320 x^{24} + 57671680 x^{19} + 69206016 x^{14} + 50331648 x^{9} + 16777216 x^{4}

= \int x^{64} + 48 x^{59} + 1056 x^{54} + 14080 x^{49} + 126720 x^{44} + 811008 x^{39} + 3784704 x^{34} + 12976128 x^{29} + 32440320 x^{24} + 57671680 x^{19} + 69206016 x^{14} + 50331648 x^{9} + 16777216 x^{4} 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 x^{64} d x + \int 48 x^{59} d x + \int 1056 x^{54} d x + \int 14080 x^{49} d x + \int 126720 x^{44} d x + \int 811008 x^{39} d x + \int 3784704 x^{34} d x + \int 12976128 x^{29} d x + \int 32440320 x^{24} d x + \int 57671680 x^{19} d x + \int 69206016 x^{14} d x + \int 50331648 x^{9} d x + \int 16777216 x^{4} d x

\int x^{64} d x = \frac{x ^{65}}{65}

\int 48 x^{59} d x = \frac{4 x ^{60}}{5}

\int 1056 x^{54} d x = \frac{96 x ^{55}}{5}

\int 14080 x^{49} d x = \frac{1408 x ^{50}}{5}

\int 126720 x^{44} d x = 2816 x^{45}

\int 811008 x^{39} d x = \frac{101376 x ^{40}}{5}

\int 3784704 x^{34} d x = \frac{540672 x ^{35}}{5}

\int 12976128 x^{29} d x = \frac{2162688 x ^{30}}{5}

\int 32440320 x^{24} d x = \frac{6488064 x ^{25}}{5}

\int 57671680 x^{19} d x = 2883584 x^{20}

\int 69206016 x^{14} d x = \frac{23068672 x ^{15}}{5}

\int 50331648 x^{9} d x = \frac{25165824 x ^{10}}{5}

\int 16777216 x^{4} d x = \frac{16777216 x ^{5}}{5}

= \frac{x ^{65}}{65} + \frac{4 x ^{60}}{5} + \frac{96 x ^{55}}{5} + \frac{1408 x ^{50}}{5} + 2816 x^{45} + \frac{101376 x ^{40}}{5} + \frac{540672 x ^{35}}{5} + \frac{2162688 x ^{30}}{5} + \frac{6488064 x ^{25}}{5} + 2883584 x^{20} + \frac{23068672 x ^{15}}{5} + \frac{25165824 x ^{10}}{5} + \frac{16777216 x ^{5}}{5}

解答补常数

= \frac{x ^{65}}{65} + \frac{4 x ^{60}}{5} + \frac{96 x ^{55}}{5} + \frac{1408 x ^{50}}{5} + 2816 x^{45} + \frac{101376 x ^{40}}{5} + \frac{540672 x ^{35}}{5} + \frac{2162688 x ^{30}}{5} + \frac{6488064 x ^{25}}{5} + 2883584 x^{20} + \frac{23068672 x ^{15}}{5} + \frac{25165824 x ^{10}}{5} + \frac{16777216 x ^{5}}{5} + C

\int \frac{x ^{2} + 1}{( x - 2 ) ^{3}} d x

\int \frac{x + 2}{x ^{2}} d x

a re a 2^{x}, - 4, 4

\int \frac{3 ^{x}}{3 - 3 ^{x}} d x

\frac{d}{d x} (\frac{ln ( x )}{sin ( x )})