integral of (x+3)(x^2+6x-7)^6

解答

\int (x + 3) (x^{2} + 6 x - 7)^{6} d x

\frac{x ^{14}}{14} + 3 x^{13} + \frac{101 x ^{12}}{2} + 414 x^{11} + \frac{2847 x ^{10}}{2} - 1251 x^{9} - \frac{38219 x ^{8}}{2} - \frac{35844 x ^{7}}{7} + \frac{267533 x ^{6}}{2} - 61299 x^{5} - \frac{976521 x ^{4}}{2} + 994014 x^{3} - \frac{1697507 x ^{2}}{2} + 352947 x + C

求解步骤

\int (x + 3) (x^{2} + 6 x - 7)^{6} d x

乘开

(x + 3) (x^{2} + 6 x - 7)^{6} : x^{13} + 39 x^{12} + 606 x^{11} + 4554 x^{10} + 14235 x^{9} - 11259 x^{8} - 152876 x^{7} - 35844 x^{6} + 802599 x^{5} - 306495 x^{4} - 1953042 x^{3} + 2982042 x^{2} - 1697507 x + 352947

= \int x^{13} + 39 x^{12} + 606 x^{11} + 4554 x^{10} + 14235 x^{9} - 11259 x^{8} - 152876 x^{7} - 35844 x^{6} + 802599 x^{5} - 306495 x^{4} - 1953042 x^{3} + 2982042 x^{2} - 1697507 x + 352947 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^{13} d x + \int 39 x^{12} d x + \int 606 x^{11} d x + \int 4554 x^{10} d x + \int 14235 x^{9} d x - \int 11259 x^{8} d x - \int 152876 x^{7} d x - \int 35844 x^{6} d x + \int 802599 x^{5} d x - \int 306495 x^{4} d x - \int 1953042 x^{3} d x + \int 2982042 x^{2} d x - \int 1697507 x d x + \int 352947 d x

\int x^{13} d x = \frac{x ^{14}}{14}

\int 39 x^{12} d x = 3 x^{13}

\int 606 x^{11} d x = \frac{101 x ^{12}}{2}

\int 4554 x^{10} d x = 414 x^{11}

\int 14235 x^{9} d x = \frac{2847 x ^{10}}{2}

\int 11259 x^{8} d x = 1251 x^{9}

\int 152876 x^{7} d x = \frac{38219 x ^{8}}{2}

\int 35844 x^{6} d x = \frac{35844 x ^{7}}{7}

\int 802599 x^{5} d x = \frac{267533 x ^{6}}{2}

\int 306495 x^{4} d x = 61299 x^{5}

\int 1953042 x^{3} d x = \frac{976521 x ^{4}}{2}

\int 2982042 x^{2} d x = 994014 x^{3}

\int 1697507 x d x = \frac{1697507 x ^{2}}{2}

\int 352947 d x = 352947 x

= \frac{x ^{14}}{14} + 3 x^{13} + \frac{101 x ^{12}}{2} + 414 x^{11} + \frac{2847 x ^{10}}{2} - 1251 x^{9} - \frac{38219 x ^{8}}{2} - \frac{35844 x ^{7}}{7} + \frac{267533 x ^{6}}{2} - 61299 x^{5} - \frac{976521 x ^{4}}{2} + 994014 x^{3} - \frac{1697507 x ^{2}}{2} + 352947 x

解答补常数

= \frac{x ^{14}}{14} + 3 x^{13} + \frac{101 x ^{12}}{2} + 414 x^{11} + \frac{2847 x ^{10}}{2} - 1251 x^{9} - \frac{38219 x ^{8}}{2} - \frac{35844 x ^{7}}{7} + \frac{267533 x ^{6}}{2} - 61299 x^{5} - \frac{976521 x ^{4}}{2} + 994014 x^{3} - \frac{1697507 x ^{2}}{2} + 352947 x + C

y^{^{'}} = y (x y^{5} + 4)

d er i v a t i v e (x^{3} + 7)^{\frac{2}{3}}

t an g e n t f (x) = - 3 x^{2} - 3 x + 2, at x = 1

d er i v a t i v e f (x) = 3 x^{2} + e^{2 x} + π^{2}

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