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受欢迎的代数 >

展开 (sinh(x)+cosh(x))^4

解答

展开

(sinh (x) + cosh (x))^{4}

解答

sinh^{4} (x) + 4 sinh^{3} (x) cosh (x) + 6 sinh^{2} (x) cosh^{2} (x) + 4 cosh^{3} (x) sinh (x) + cosh^{4} (x)

求解步骤

(sinh (x) + cosh (x))^{4}

使用二项式定理:

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

a = sinh (x), b = cosh (x)

= i = 0 \sum 4 (i 4) sinh^{(4 - i)} (x) cosh^{i} (x)

展开求和

= \frac{4 !}{0 ! ( 4 - 0 ) !} sinh^{4} (x) cosh^{0} (x) + \frac{4 !}{1 ! ( 4 - 1 ) !} sinh^{3} (x) cosh^{1} (x) + \frac{4 !}{2 ! ( 4 - 2 ) !} sinh^{2} (x) cosh^{2} (x) + \frac{4 !}{3 ! ( 4 - 3 ) !} sinh^{1} (x) cosh^{3} (x) + \frac{4 !}{4 ! ( 4 - 4 ) !} sinh^{0} (x) cosh^{4} (x)

化简

\frac{4 !}{0 ! ( 4 - 0 ) !} sinh^{4} (x) cosh^{0} (x) : sinh^{4} (x)

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} sinh^{3} (x) cosh^{1} (x) : 4 sinh^{3} (x) cosh (x)

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} sinh^{2} (x) cosh^{2} (x) : 6 sinh^{2} (x) cosh^{2} (x)

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} sinh^{1} (x) cosh^{3} (x) : 4 cosh^{3} (x) sinh (x)

化简

\frac{4 !}{4 ! ( 4 - 4 ) !} sinh^{0} (x) cosh^{4} (x) : cosh^{4} (x)

= sinh^{4} (x) + 4 sinh^{3} (x) cosh (x) + 6 sinh^{2} (x) cosh^{2} (x) + 4 cosh^{3} (x) sinh (x) + cosh^{4} (x)

流行的例子

展开 1/((s+4)^2)

e x p an d \frac{1}{( s + 4 ) ^{2}}

展开 (s+3)/(s(s+2)(s+1))

e x p an d \frac{s + 3}{s ( s + 2 ) ( s + 1 )}

化简 \delta(x-4)

s im pl i f y δ (x - 4)

展开 (1/2 x^3+6x)^2

e x p an d (\frac{1}{2} x^{3} + 6 x)^{2}

化简 (5sqrt(x)+sqrt(2))(5sqrt(x)-sqrt(2))

s im pl i f y (5 x + 2) (5 x - 2)