erweitern (1/(x^2)+2x)^5

Lösung

erweitern

(\frac{1}{x ^{2}} + 2 x)^{5}

\frac{1}{x ^{10}} + \frac{10}{x ^{7}} + \frac{40}{x ^{4}} + \frac{80}{x} + 80 x^{2} + 32 x^{5}

Schritte zur Lösung

(\frac{1}{x ^{2}} + 2 x)^{5}

Wende Binomialsetz an:

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

a = \frac{1}{x ^{2}}, b = 2 x

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

Erweitere Summation

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

Vereinfache

\frac{5 !}{0 ! ( 5 - 0 ) !} (\frac{1}{x ^{2}})^{5} (2 x)^{0} : \frac{1}{x ^{10}}

Vereinfache

\frac{5 !}{1 ! ( 5 - 1 ) !} (\frac{1}{x ^{2}})^{4} (2 x)^{1} : \frac{10}{x ^{7}}

Vereinfache

\frac{5 !}{2 ! ( 5 - 2 ) !} (\frac{1}{x ^{2}})^{3} (2 x)^{2} : \frac{40}{x ^{4}}

Vereinfache

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

Vereinfache

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

Vereinfache

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

= \frac{1}{x ^{10}} + \frac{10}{x ^{7}} + \frac{40}{x ^{4}} + \frac{80}{x} + 80 x^{2} + 32 x^{5}

e x p an d \frac{( x - 4 ) ^{2}}{x ^{2} - 9}

e x p an d (15 x^{4} - 3 x^{2} + 1) cos (x)

e x p an d π (- \frac{x ^{5}}{5} + \frac{26 x ^{3}}{3} - 25 x) + C

e x p an d \frac{10 ( a - b )}{a + b}

s im pl i f y (1 - lo g_{2} (12)) \cdot (1 - lo g_{6} (12))