泰勒 (2x)/(1+x^2)

解答

泰勒

\frac{2 x}{1 + x ^{2}}

2 x - 2 x^{3} + 2 x^{5} - 2 x^{7} + 2 x^{9} + \dots

求解步骤

应用泰勒公式

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

求导数

= 0 + \frac{2}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 12}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{240}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- 10080}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{725760}{9 !} x^{9} + \dots

整理后得

= 2 x - 2 x^{3} + 2 x^{5} - 2 x^{7} + 2 x^{9} + \dots

\int \frac{x + 9}{2 x ^{3} - 18 x} d x

2 \frac{d y}{d x} + 18 y = 9

\int 5 sin^{2} (3 x) d x

\int_{0}^{1} \frac{1}{x ^{0.2}} d x

x \to 0 - lim (x (x - 1))