integral of 7x^6(x^7+1)^8

解答

\int 7 x^{6} (x^{7} + 1)^{8} d x

\frac{x ^{63}}{9} + x^{56} + 4 x^{49} + \frac{28 x ^{42}}{3} + 14 x^{35} + 14 x^{28} + \frac{28 x ^{21}}{3} + 4 x^{14} + x^{7} + C

求解步骤

\int 7 x^{6} (x^{7} + 1)^{8} d x

提出常数:

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

= 7 \cdot \int x^{6} (x^{7} + 1)^{8} d x

乘开

x^{6} (x^{7} + 1)^{8} : x^{62} + 8 x^{55} + 28 x^{48} + 56 x^{41} + 70 x^{34} + 56 x^{27} + 28 x^{20} + 8 x^{13} + x^{6}

= 7 \cdot \int x^{62} + 8 x^{55} + 28 x^{48} + 56 x^{41} + 70 x^{34} + 56 x^{27} + 28 x^{20} + 8 x^{13} + 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

= 7 (\int x^{62} d x + \int 8 x^{55} d x + \int 28 x^{48} d x + \int 56 x^{41} d x + \int 70 x^{34} d x + \int 56 x^{27} d x + \int 28 x^{20} d x + \int 8 x^{13} d x + \int x^{6} d x)

\int x^{62} d x = \frac{x ^{63}}{63}

\int 8 x^{55} d x = \frac{x ^{56}}{7}

\int 28 x^{48} d x = \frac{4 x ^{49}}{7}

\int 56 x^{41} d x = \frac{4 x ^{42}}{3}

\int 70 x^{34} d x = 2 x^{35}

\int 56 x^{27} d x = 2 x^{28}

\int 28 x^{20} d x = \frac{4 x ^{21}}{3}

\int 8 x^{13} d x = \frac{4 x ^{14}}{7}

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

= 7 (\frac{x ^{63}}{63} + \frac{x ^{56}}{7} + \frac{4 x ^{49}}{7} + \frac{4 x ^{42}}{3} + 2 x^{35} + 2 x^{28} + \frac{4 x ^{21}}{3} + \frac{4 x ^{14}}{7} + \frac{x ^{7}}{7})

化简

7 (\frac{x ^{63}}{63} + \frac{x ^{56}}{7} + \frac{4 x ^{49}}{7} + \frac{4 x ^{42}}{3} + 2 x^{35} + 2 x^{28} + \frac{4 x ^{21}}{3} + \frac{4 x ^{14}}{7} + \frac{x ^{7}}{7}) : \frac{x ^{63}}{9} + x^{56} + 4 x^{49} + \frac{28 x ^{42}}{3} + 14 x^{35} + 14 x^{28} + \frac{28 x ^{21}}{3} + 4 x^{14} + x^{7}

= \frac{x ^{63}}{9} + x^{56} + 4 x^{49} + \frac{28 x ^{42}}{3} + 14 x^{35} + 14 x^{28} + \frac{28 x ^{21}}{3} + 4 x^{14} + x^{7}

解答补常数

= \frac{x ^{63}}{9} + x^{56} + 4 x^{49} + \frac{28 x ^{42}}{3} + 14 x^{35} + 14 x^{28} + \frac{28 x ^{21}}{3} + 4 x^{14} + x^{7} + C

\int x^{2} 25 + x^{2} d x

\int x^{- 4} (x - 2)^{2} d x

\int - \frac{2}{81 - 49 x ^{2}} d x

\int \frac{3 x ^{2}}{( 49 + x ^{2} ) ^{2}} d x

\int a^{k x} d x