integral of (2x^2+4)^8

解答

\int (2 x^{2} + 4)^{8} d x

\frac{256 x ^{17}}{17} + \frac{4096 x ^{15}}{15} + \frac{28672 x ^{13}}{13} + \frac{114688 x ^{11}}{11} + \frac{286720 x ^{9}}{9} + 65536 x^{7} + \frac{458752 x ^{5}}{5} + \frac{262144 x ^{3}}{3} + 65536 x + C

求解步骤

\int (2 x^{2} + 4)^{8} d x

乘开

(2 x^{2} + 4)^{8} : 256 x^{16} + 4096 x^{14} + 28672 x^{12} + 114688 x^{10} + 286720 x^{8} + 458752 x^{6} + 458752 x^{4} + 262144 x^{2} + 65536

= \int 256 x^{16} + 4096 x^{14} + 28672 x^{12} + 114688 x^{10} + 286720 x^{8} + 458752 x^{6} + 458752 x^{4} + 262144 x^{2} + 65536 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 256 x^{16} d x + \int 4096 x^{14} d x + \int 28672 x^{12} d x + \int 114688 x^{10} d x + \int 286720 x^{8} d x + \int 458752 x^{6} d x + \int 458752 x^{4} d x + \int 262144 x^{2} d x + \int 65536 d x

\int 256 x^{16} d x = \frac{256 x ^{17}}{17}

\int 4096 x^{14} d x = \frac{4096 x ^{15}}{15}

\int 28672 x^{12} d x = \frac{28672 x ^{13}}{13}

\int 114688 x^{10} d x = \frac{114688 x ^{11}}{11}

\int 286720 x^{8} d x = \frac{286720 x ^{9}}{9}

\int 458752 x^{6} d x = 65536 x^{7}

\int 458752 x^{4} d x = \frac{458752 x ^{5}}{5}

\int 262144 x^{2} d x = \frac{262144 x ^{3}}{3}

\int 65536 d x = 65536 x

= \frac{256 x ^{17}}{17} + \frac{4096 x ^{15}}{15} + \frac{28672 x ^{13}}{13} + \frac{114688 x ^{11}}{11} + \frac{286720 x ^{9}}{9} + 65536 x^{7} + \frac{458752 x ^{5}}{5} + \frac{262144 x ^{3}}{3} + 65536 x

解答补常数

= \frac{256 x ^{17}}{17} + \frac{4096 x ^{15}}{15} + \frac{28672 x ^{13}}{13} + \frac{114688 x ^{11}}{11} + \frac{286720 x ^{9}}{9} + 65536 x^{7} + \frac{458752 x ^{5}}{5} + \frac{262144 x ^{3}}{3} + 65536 x + C

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