erweitern (sqrt(log_{10)(x)}-1/(ln^2(2x)))^3

Lösung

erweitern

(lo g_{10} (x) - \frac{1}{ln ^{2} ( 2 x )})^{3}

lo g_{10} (x)^{\frac{3}{2}} - \frac{3 lo g _{10} ( x )}{ln ^{2} ( 2 x )} + \frac{3 lo g _{10} ( x )}{ln ^{4} ( 2 x )} - \frac{1}{ln ^{6} ( 2 x )}

Schritte zur Lösung

(lo g_{10} (x) - \frac{1}{ln ^{2} ( 2 x )})^{3}

Wende Formel für perfekte dritte Potenzen an:

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

a = lo g_{10} (x), b = \frac{1}{ln ^{2} ( 2 x )}

= (lo g_{10} (x))^{3} - 3 (lo g_{10} (x))^{2} \frac{1}{ln ^{2} ( 2 x )} + 3 lo g_{10} (x) (\frac{1}{ln ^{2} ( 2 x )})^{2} - (\frac{1}{ln ^{2} ( 2 x )})^{3}

Vereinfache

(lo g_{10} (x))^{3} - 3 (lo g_{10} (x))^{2} \frac{1}{ln ^{2} ( 2 x )} + 3 lo g_{10} (x) (\frac{1}{ln ^{2} ( 2 x )})^{2} - (\frac{1}{ln ^{2} ( 2 x )})^{3} : lo g_{10} (x)^{\frac{3}{2}} - \frac{3 lo g _{10} ( x )}{ln ^{2} ( 2 x )} + \frac{3 lo g _{10} ( x )}{ln ^{4} ( 2 x )} - \frac{1}{ln ^{6} ( 2 x )}

= lo g_{10} (x)^{\frac{3}{2}} - \frac{3 lo g _{10} ( x )}{ln ^{2} ( 2 x )} + \frac{3 lo g _{10} ( x )}{ln ^{4} ( 2 x )} - \frac{1}{ln ^{6} ( 2 x )}

e x p an d \frac{e ^{- s - 1} + 4}{( s + 5 ) ( s + 1 )}

e x p an d x^{2} (5 + 2 x^{3})^{8} d x

e x p an d \frac{( 2 s ^{2} + 7 s + 7 )}{( s + 2 ) ( s + 1 )}

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

e x p an d \frac{- x ^{2} + 4 x + 12}{x ( x - 4 )}