erweitern (cos^{210}(x)+sin^{210}(x))^5

Lösung

erweitern

(cos^{210} (x) + sin^{210} (x))^{5}

cos^{1050} (x) + 5 cos^{840} (x) sin^{210} (x) + 10 cos^{630} (x) sin^{420} (x) + 10 sin^{630} (x) cos^{420} (x) + 5 sin^{840} (x) cos^{210} (x) + sin^{1050} (x)

Schritte zur Lösung

(cos^{210} (x) + sin^{210} (x))^{5}

Wende Binomialsetz an:

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

a = cos^{210} (x), b = sin^{210} (x)

= i = 0 \sum 5 (i 5) (cos^{210} (x))^{(5 - i)} (sin^{210} (x))^{i}

Erweitere Summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} (cos^{210} (x))^{5} (sin^{210} (x))^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} (cos^{210} (x))^{4} (sin^{210} (x))^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} (cos^{210} (x))^{3} (sin^{210} (x))^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} (cos^{210} (x))^{2} (sin^{210} (x))^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} (cos^{210} (x))^{1} (sin^{210} (x))^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} (cos^{210} (x))^{0} (sin^{210} (x))^{5}

Vereinfache

\frac{5 !}{0 ! ( 5 - 0 ) !} (cos^{210} (x))^{5} (sin^{210} (x))^{0} : cos^{1050} (x)

Vereinfache

\frac{5 !}{1 ! ( 5 - 1 ) !} (cos^{210} (x))^{4} (sin^{210} (x))^{1} : 5 cos^{840} (x) sin^{210} (x)

Vereinfache

\frac{5 !}{2 ! ( 5 - 2 ) !} (cos^{210} (x))^{3} (sin^{210} (x))^{2} : 10 cos^{630} (x) sin^{420} (x)

Vereinfache

\frac{5 !}{3 ! ( 5 - 3 ) !} (cos^{210} (x))^{2} (sin^{210} (x))^{3} : 10 sin^{630} (x) cos^{420} (x)

Vereinfache

\frac{5 !}{4 ! ( 5 - 4 ) !} (cos^{210} (x))^{1} (sin^{210} (x))^{4} : 5 sin^{840} (x) cos^{210} (x)

Vereinfache

\frac{5 !}{5 ! ( 5 - 5 ) !} (cos^{210} (x))^{0} (sin^{210} (x))^{5} : sin^{1050} (x)

= cos^{1050} (x) + 5 cos^{840} (x) sin^{210} (x) + 10 cos^{630} (x) sin^{420} (x) + 10 sin^{630} (x) cos^{420} (x) + 5 sin^{840} (x) cos^{210} (x) + sin^{1050} (x)

s im pl i f y (x - 1) \cdot (2 x + 5)

s im pl i f y (3 - 3 i)^{5}

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

\frac{d}{d x} ((y^{2}) (5 sin (x) + 5 cos (x)))

s im pl i f y (s^{2} + 4) \cdot (s^{2} + \frac{4}{3})