taylor 1/(1+sin(pi*x))

Lösung

taylor

\frac{1}{1 + sin ( π \cdot x )}

1 - π x + π^{2} x^{2} - \frac{5 π ^{3}}{6} x^{3} + \frac{2 π ^{4}}{3} x^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( \frac{1}{1 + s i n ( π x )} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{1 + s i n ( π x )} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{1 + s i n ( π x )} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 1 + \frac{- π}{1 !} x + \frac{2 π ^{2}}{2 !} x^{2} + \frac{- 5 π ^{3}}{3 !} x^{3} + \frac{16 π ^{4}}{4 !} x^{4} + \dots

Fasse zusammen

= 1 - π x + π^{2} x^{2} - \frac{5 π ^{3}}{6} x^{3} + \frac{2 π ^{4}}{3} x^{4} + \dots

\frac{d}{d x} \frac{1}{3 ( 6 x ^{2} + 5 ) ^{2}}

\int \frac{k}{x ln ( x )} d x

\frac{d}{d x} (\frac{x}{x ^{2} - 8})

\frac{d}{d x} (y z e^{x})

\frac{d y}{d x} = x y + 2 x + 3 y + 6