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受欢迎的代数 >

展开 (1+2sin(x))^5

解答

展开

(1 + 2 sin (x))^{5}

解答

1 + 10 sin (x) + 40 sin^{2} (x) + 80 sin^{3} (x) + 80 sin^{4} (x) + 32 sin^{5} (x)

求解步骤

(1 + 2 sin (x))^{5}

使用二项式定理:

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

a = 1, b = 2 sin (x)

= i = 0 \sum 5 (i 5) \cdot 1^{(5 - i)} (2 sin (x))^{i}

展开求和

= \frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (2 sin (x))^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (2 sin (x))^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (2 sin (x))^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (2 sin (x))^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (2 sin (x))^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (2 sin (x))^{5}

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (2 sin (x))^{0} : 1

化简

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (2 sin (x))^{1} : 10 sin (x)

化简

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (2 sin (x))^{2} : 40 sin^{2} (x)

化简

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (2 sin (x))^{3} : 80 sin^{3} (x)

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (2 sin (x))^{4} : 80 sin^{4} (x)

化简

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (2 sin (x))^{5} : 32 sin^{5} (x)

= 1 + 10 sin (x) + 40 sin^{2} (x) + 80 sin^{3} (x) + 80 sin^{4} (x) + 32 sin^{5} (x)

流行的例子

展开 (1+2sin(x))^2

e x p an d (1 + 2 sin (x))^{2}

展开 (6e^x(-e^x+6))/((6+e^x)^3)

e x p an d \frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}

展开 (2,1+i,i)(2-i,2,1+2i)

e x p an d (2, 1 + i, i) (2 - i, 2, 1 + 2 i)

展开-(2(2x-1))/(3(x^2-x)^{1/3)}

e x p an d - \frac{2 ( 2 x - 1 )}{3 ( x ^{2} - x ) ^{\frac{1}{3}}}

展开 (x^{-1})^2

e x p an d (x^{- 1})^{2}