espandere (0.3+0.7e^x)^5

Soluzione

espandere

(0.3 + 0.7 e^{x})^{5}

0.00243 + 0.02835 e^{x} + 0.1323 e^{2 x} + 0.3087 e^{3 x} + 0.36015 e^{4 x} + 0.16807 e^{5 x}

Fasi della soluzione

(0.3 + 0.7 e^{x})^{5}

Applicare il teorema binomiale:

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

a = 0.3, b = 0.7 e^{x}

= i = 0 \sum 5 (i 5) \cdot 0. 3^{(5 - i)} (0.7 e^{x})^{i}

Espandere la sommatoria

= \frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 0. 3^{5} (0.7 e^{x})^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 0. 3^{4} (0.7 e^{x})^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 0. 3^{3} (0.7 e^{x})^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 0. 3^{2} (0.7 e^{x})^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 0. 3^{1} (0.7 e^{x})^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 0. 3^{0} (0.7 e^{x})^{5}

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 0. 3^{5} (0.7 e^{x})^{0} = 0.00243

Semplifica

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 0. 3^{4} (0.7 e^{x})^{1} : 0.02835 e^{x}

Semplifica

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 0. 3^{3} (0.7 e^{x})^{2} : 0.1323 e^{2 x}

Semplifica

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 0. 3^{2} (0.7 e^{x})^{3} : 0.3087 e^{3 x}

Semplifica

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 0. 3^{1} (0.7 e^{x})^{4} : 0.36015 e^{4 x}

Semplifica

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 0. 3^{0} (0.7 e^{x})^{5} : 0.16807 e^{5 x}

= 0.00243 + 0.02835 e^{x} + 0.1323 e^{2 x} + 0.3087 e^{3 x} + 0.36015 e^{4 x} + 0.16807 e^{5 x}

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

e x p an d \frac{( x y - 2 y ) ^{6}}{( x - 2 ) ^{6} + 4 y ^{6}}

e x p an d 2 L (x) x + L (x) (x^{2} + 2 = L (x) (x)^{3})

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

e x p an d (3 d - 5 e) (9 d^{2} + 15 d e + 25 e^{2})