展开 x(2x+1)^{10}dx

解答

展开

x (2 x + 1)^{10} d x

1024 x^{11} d x + 5120 x^{10} d x + 11520 x^{9} d x + 15360 x^{8} d x + 13440 x^{7} d x + 8064 x^{6} d x + 3360 x^{5} d x + 960 x^{4} d x + 180 x^{3} d x + 20 x^{2} d x + x d x

求解步骤

x (2 x + 1)^{10} d x

(2 x + 1)^{10} = 1024 x^{10} + 5120 x^{9} + 11520 x^{8} + 15360 x^{7} + 13440 x^{6} + 8064 x^{5} + 3360 x^{4} + 960 x^{3} + 180 x^{2} + 20 x + 1

= x (1024 x^{10} + 5120 x^{9} + 11520 x^{8} + 15360 x^{7} + 13440 x^{6} + 8064 x^{5} + 3360 x^{4} + 960 x^{3} + 180 x^{2} + 20 x + 1) d x

打开括号

= x d x \cdot 1024 x^{10} + x d x \cdot 5120 x^{9} + x d x \cdot 11520 x^{8} + x d x \cdot 15360 x^{7} + x d x \cdot 13440 x^{6} + x d x \cdot 8064 x^{5} + x d x \cdot 3360 x^{4} + x d x \cdot 960 x^{3} + x d x \cdot 180 x^{2} + x d x \cdot 20 x + x d x \cdot 1

= 1024 x^{10} x d x + 5120 x^{9} x d x + 11520 x^{8} x d x + 15360 x^{7} x d x + 13440 x^{6} x d x + 8064 x^{5} x d x + 3360 x^{4} x d x + 960 x^{3} x d x + 180 x^{2} x d x + 20 xx d x + 1 \cdot x d x

化简

1024 x^{10} x d x + 5120 x^{9} x d x + 11520 x^{8} x d x + 15360 x^{7} x d x + 13440 x^{6} x d x + 8064 x^{5} x d x + 3360 x^{4} x d x + 960 x^{3} x d x + 180 x^{2} x d x + 20 xx d x + 1 \cdot x d x : 1024 x^{11} d x + 5120 x^{10} d x + 11520 x^{9} d x + 15360 x^{8} d x + 13440 x^{7} d x + 8064 x^{6} d x + 3360 x^{5} d x + 960 x^{4} d x + 180 x^{3} d x + 20 x^{2} d x + x d x

= 1024 x^{11} d x + 5120 x^{10} d x + 11520 x^{9} d x + 15360 x^{8} d x + 13440 x^{7} d x + 8064 x^{6} d x + 3360 x^{5} d x + 960 x^{4} d x + 180 x^{3} d x + 20 x^{2} d x + x d x

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