integral of 5x^6(2x^7+5)^4

解答

\int 5 x^{6} (2 x^{7} + 5)^{4} d x

5 (\frac{16 x ^{35}}{35} + \frac{40 x ^{28}}{7} + \frac{200 x ^{21}}{7} + \frac{500 x ^{14}}{7} + \frac{625 x ^{7}}{7}) + C

求解步骤

\int 5 x^{6} (2 x^{7} + 5)^{4} d x

提出常数:

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

= 5 \cdot \int x^{6} (2 x^{7} + 5)^{4} d x

乘开

x^{6} (2 x^{7} + 5)^{4} : 16 x^{34} + 160 x^{27} + 600 x^{20} + 1000 x^{13} + 625 x^{6}

= 5 \cdot \int 16 x^{34} + 160 x^{27} + 600 x^{20} + 1000 x^{13} + 625 x^{6} 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 16 x^{34} d x + \int 160 x^{27} d x + \int 600 x^{20} d x + \int 1000 x^{13} d x + \int 625 x^{6} d x)

\int 16 x^{34} d x = \frac{16 x ^{35}}{35}

\int 160 x^{27} d x = \frac{40 x ^{28}}{7}

\int 600 x^{20} d x = \frac{200 x ^{21}}{7}

\int 1000 x^{13} d x = \frac{500 x ^{14}}{7}

\int 625 x^{6} d x = \frac{625 x ^{7}}{7}

= 5 (\frac{16 x ^{35}}{35} + \frac{40 x ^{28}}{7} + \frac{200 x ^{21}}{7} + \frac{500 x ^{14}}{7} + \frac{625 x ^{7}}{7})

解答补常数

= 5 (\frac{16 x ^{35}}{35} + \frac{40 x ^{28}}{7} + \frac{200 x ^{21}}{7} + \frac{500 x ^{14}}{7} + \frac{625 x ^{7}}{7}) + C

a re a x^{3} + 3 x^{2} + 2 x, - 3, 3

e^{x}, 2 x - 1, x = - 1, x = 2

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

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

\frac{d ^{2}}{d x ^{2}} (e^{- 3 x^{2}})