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受欢迎的代数 >

展开 (e^{3x}+sin(2x))^4

解答

展开

(e^{3 x} + sin (2 x))^{4}

解答

e^{12 x} + 4 e^{9 x} sin (2 x) + 6 e^{6 x} sin^{2} (2 x) + 4 e^{3 x} sin^{3} (2 x) + sin^{4} (2 x)

求解步骤

(e^{3 x} + sin (2 x))^{4}

使用二项式定理:

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

a = e^{3 x}, b = sin (2 x)

= i = 0 \sum 4 (i 4) (e^{3 x})^{(4 - i)} sin^{i} (2 x)

展开求和

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

化简

\frac{4 !}{0 ! ( 4 - 0 ) !} (e^{3 x})^{4} sin^{0} (2 x) : e^{12 x}

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} (e^{3 x})^{3} sin^{1} (2 x) : 4 e^{9 x} sin (2 x)

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} (e^{3 x})^{2} sin^{2} (2 x) : 6 e^{6 x} sin^{2} (2 x)

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} (e^{3 x})^{1} sin^{3} (2 x) : 4 e^{3 x} sin^{3} (2 x)

化简

\frac{4 !}{4 ! ( 4 - 4 ) !} (e^{3 x})^{0} sin^{4} (2 x) : sin^{4} (2 x)

= e^{12 x} + 4 e^{9 x} sin (2 x) + 6 e^{6 x} sin^{2} (2 x) + 4 e^{3 x} sin^{3} (2 x) + sin^{4} (2 x)

流行的例子

展开 3(6ln(3x+7)+9)

e x p an d 3 (6 ln (3 x + 7) + 9)

展开 x^{4/5}(x-5)

e x p an d x^{\frac{4}{5}} (x - 5)

展开 x^{4/5}(x-4)

e x p an d x^{\frac{4}{5}} (x - 4)

展开 (x+3)/(x(x-9))

e x p an d \frac{x + 3}{x ( x - 9 )}

展开 (x^2+3x+2)/((3x+1)^2)

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