integral of 6x^3(-3x^4+3)^6

解答

\int 6 x^{3} (- 3 x^{4} + 3)^{6} d x

6 (\frac{729 x ^{28}}{28} - \frac{729 x ^{24}}{4} + \frac{2187 x ^{20}}{4} - \frac{3645 x ^{16}}{4} + \frac{3645 x ^{12}}{4} - \frac{2187 x ^{8}}{4} + \frac{729 x ^{4}}{4}) + C

求解步骤

\int 6 x^{3} (- 3 x^{4} + 3)^{6} d x

提出常数:

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

= 6 \cdot \int x^{3} (- 3 x^{4} + 3)^{6} d x

乘开

x^{3} (- 3 x^{4} + 3)^{6} : 729 x^{27} - 4374 x^{23} + 10935 x^{19} - 14580 x^{15} + 10935 x^{11} - 4374 x^{7} + 729 x^{3}

= 6 \cdot \int 729 x^{27} - 4374 x^{23} + 10935 x^{19} - 14580 x^{15} + 10935 x^{11} - 4374 x^{7} + 729 x^{3} 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

= 6 (\int 729 x^{27} d x - \int 4374 x^{23} d x + \int 10935 x^{19} d x - \int 14580 x^{15} d x + \int 10935 x^{11} d x - \int 4374 x^{7} d x + \int 729 x^{3} d x)

\int 729 x^{27} d x = \frac{729 x ^{28}}{28}

\int 4374 x^{23} d x = \frac{729 x ^{24}}{4}

\int 10935 x^{19} d x = \frac{2187 x ^{20}}{4}

\int 14580 x^{15} d x = \frac{3645 x ^{16}}{4}

\int 10935 x^{11} d x = \frac{3645 x ^{12}}{4}

\int 4374 x^{7} d x = \frac{2187 x ^{8}}{4}

\int 729 x^{3} d x = \frac{729 x ^{4}}{4}

= 6 (\frac{729 x ^{28}}{28} - \frac{729 x ^{24}}{4} + \frac{2187 x ^{20}}{4} - \frac{3645 x ^{16}}{4} + \frac{3645 x ^{12}}{4} - \frac{2187 x ^{8}}{4} + \frac{729 x ^{4}}{4})

解答补常数

= 6 (\frac{729 x ^{28}}{28} - \frac{729 x ^{24}}{4} + \frac{2187 x ^{20}}{4} - \frac{3645 x ^{16}}{4} + \frac{3645 x ^{12}}{4} - \frac{2187 x ^{8}}{4} + \frac{729 x ^{4}}{4}) + C

\int \frac{1}{x + x ^{2}} d x

\int \frac{cos ( x )}{sin ^{3} ( x ) + sin ( x )} d x

\int \frac{x ^{2} - 1}{2 x ^{2} - 32} d x

\int \frac{x ^{2} + 7}{x ^{2} + 9} d x

\int (6 x + 3) e^{- 2 x} d x