integral of x(4+x)^{10}

解答

\int x (4 + x)^{10} d x

524288 x^{2} + \frac{2621440 x ^{3}}{3} + 737280 x^{4} + 393216 x^{5} + 143360 x^{6} + 36864 x^{7} + 6720 x^{8} + \frac{2560 x ^{9}}{3} + 72 x^{10} + \frac{40 x ^{11}}{11} + \frac{x ^{12}}{12} + C

求解步骤

\int x (4 + x)^{10} d x

乘开

x (4 + x)^{10} : 1048576 x + 2621440 x^{2} + 2949120 x^{3} + 1966080 x^{4} + 860160 x^{5} + 258048 x^{6} + 53760 x^{7} + 7680 x^{8} + 720 x^{9} + 40 x^{10} + x^{11}

= \int 1048576 x + 2621440 x^{2} + 2949120 x^{3} + 1966080 x^{4} + 860160 x^{5} + 258048 x^{6} + 53760 x^{7} + 7680 x^{8} + 720 x^{9} + 40 x^{10} + x^{11} 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 1048576 x d x + \int 2621440 x^{2} d x + \int 2949120 x^{3} d x + \int 1966080 x^{4} d x + \int 860160 x^{5} d x + \int 258048 x^{6} d x + \int 53760 x^{7} d x + \int 7680 x^{8} d x + \int 720 x^{9} d x + \int 40 x^{10} d x + \int x^{11} d x

\int 1048576 x d x = 524288 x^{2}

\int 2621440 x^{2} d x = \frac{2621440 x ^{3}}{3}

\int 2949120 x^{3} d x = 737280 x^{4}

\int 1966080 x^{4} d x = 393216 x^{5}

\int 860160 x^{5} d x = 143360 x^{6}

\int 258048 x^{6} d x = 36864 x^{7}

\int 53760 x^{7} d x = 6720 x^{8}

\int 7680 x^{8} d x = \frac{2560 x ^{9}}{3}

\int 720 x^{9} d x = 72 x^{10}

\int 40 x^{10} d x = \frac{40 x ^{11}}{11}

\int x^{11} d x = \frac{x ^{12}}{12}

= 524288 x^{2} + \frac{2621440 x ^{3}}{3} + 737280 x^{4} + 393216 x^{5} + 143360 x^{6} + 36864 x^{7} + 6720 x^{8} + \frac{2560 x ^{9}}{3} + 72 x^{10} + \frac{40 x ^{11}}{11} + \frac{x ^{12}}{12}

解答补常数

= 524288 x^{2} + \frac{2621440 x ^{3}}{3} + 737280 x^{4} + 393216 x^{5} + 143360 x^{6} + 36864 x^{7} + 6720 x^{8} + \frac{2560 x ^{9}}{3} + 72 x^{10} + \frac{40 x ^{11}}{11} + \frac{x ^{12}}{12} + C

d er i v a t i v e \frac{2 ( 1 - ln ( x ) )}{x ^{2}}

x \to - 2 lim (3 x^{2} + 5 x - 1)

\int \frac{t}{( t ^{2} + 1 ) ^{\frac{5}{2}}} d t

\int \frac{1}{16 x ^{2} + 7} d x

d er i v a t i v e \frac{13}{x}