erweitern (sqrt(x+1)(2-x)^5)/((x+3)^7)

Lösung

erweitern

\frac{x + 1 ( 2 - x ) ^{5}}{( x + 3 ) ^{7}}

\frac{32 x + 1 - 80 x x + 1 + 80 x ^{2} x + 1 - 40 x ^{3} x + 1 + 10 x ^{4} x + 1 - x ^{5} x + 1}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187}

Schritte zur Lösung

\frac{x + 1 ( 2 - x ) ^{5}}{( x + 3 ) ^{7}}

(x + 3)^{7} = x^{7} + 21 x^{6} + 189 x^{5} + 945 x^{4} + 2835 x^{3} + 5103 x^{2} + 5103 x + 2187

= \frac{( - x + 2 ) ^{5} x + 1}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187}

(2 - x)^{5} = 32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} - x^{5}

= \frac{( - x ^{5} + 10 x ^{4} - 40 x ^{3} + 80 x ^{2} - 80 x + 32 ) x + 1}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187}

Multipliziere aus

x + 1 (32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} - x^{5}) : 32 x + 1 - 80 x x + 1 + 80 x^{2} x + 1 - 40 x^{3} x + 1 + 10 x^{4} x + 1 - x^{5} x + 1

= \frac{32 x + 1 - 80 x x + 1 + 80 x ^{2} x + 1 - 40 x ^{3} x + 1 + 10 x ^{4} x + 1 - x ^{5} x + 1}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187}

e x p an d 2 l e f t (1 - cos (2 x)) (1 + cot (2 x))

s im pl i f y (1 - cos (θ)) (cos (θ))

e x p an d \frac{1}{s ( s + 1 ) ( s + 10 )}

e x p an d \frac{( 1.5 ) ( 1 + 15 t ^{2} ) ^{2}}{t}

e x p an d sec (x) (1 - sin (x)) (sec (x) + tan (x))