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受欢迎的代数 >

展开 (1-cos(x))^8

解答

展开

(1 - cos (x))^{8}

解答

1 - 8 cos (x) + 28 cos^{2} (x) - 56 cos^{3} (x) + 70 cos^{4} (x) - 56 cos^{5} (x) + 28 cos^{6} (x) - 8 cos^{7} (x) + cos^{8} (x)

求解步骤

(1 - cos (x))^{8}

使用二项式定理:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 1, b = - cos (x)

= i = 0 \sum 8 (i 8) \cdot 1^{(8 - i)} (- cos (x))^{i}

展开求和

= \frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 1^{8} (- cos (x))^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 1^{7} (- cos (x))^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 1^{6} (- cos (x))^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 1^{5} (- cos (x))^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 1^{4} (- cos (x))^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 1^{3} (- cos (x))^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 1^{2} (- cos (x))^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 1^{1} (- cos (x))^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 1^{0} (- cos (x))^{8}

化简

\frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 1^{8} (- cos (x))^{0} : 1

化简

\frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 1^{7} (- cos (x))^{1} : - 8 cos (x)

化简

\frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 1^{6} (- cos (x))^{2} : 28 cos^{2} (x)

化简

\frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 1^{5} (- cos (x))^{3} : - 56 cos^{3} (x)

化简

\frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 1^{4} (- cos (x))^{4} : 70 cos^{4} (x)

化简

\frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 1^{3} (- cos (x))^{5} : - 56 cos^{5} (x)

化简

\frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 1^{2} (- cos (x))^{6} : 28 cos^{6} (x)

化简

\frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 1^{1} (- cos (x))^{7} : - 8 cos^{7} (x)

化简

\frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 1^{0} (- cos (x))^{8} : cos^{8} (x)

= 1 - 8 cos (x) + 28 cos^{2} (x) - 56 cos^{3} (x) + 70 cos^{4} (x) - 56 cos^{5} (x) + 28 cos^{6} (x) - 8 cos^{7} (x) + cos^{8} (x)

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e x p an d \frac{19}{( x + 2 ) ( x - 7 )}

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e x p an d (3 x^{2} e^{y} + x) d x + (x^{3} + 2) ey d y