麦克劳林 (x^2)/((4+x^2)^2)

解答

麦克劳林

\frac{x ^{2}}{( 4 + x ^{2} ) ^{2}}

\frac{1}{16} x^{2} - \frac{1}{32} x^{4} + \frac{3}{256} x^{6} - \frac{1}{256} x^{8} + \frac{5}{4096} x^{10} + \dots

求解步骤

应用麦克劳林公式

= 0 + \frac{\frac{d}{d x} ( \frac{x ^{2}}{( 4 + x ^{2} ) ^{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{x ^{2}}{( 4 + x ^{2} ) ^{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{x ^{2}}{( 4 + x ^{2} ) ^{2}} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 0 + \frac{0}{1 !} x + \frac{\frac{1}{8}}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{- \frac{3}{4}}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{\frac{135}{16}}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{- \frac{315}{2}}{8 !} x^{8} + \frac{0}{9 !} x^{9} + \frac{\frac{70875}{16}}{10 !} x^{10} + \dots

整理后得

= \frac{1}{16} x^{2} - \frac{1}{32} x^{4} + \frac{3}{256} x^{6} - \frac{1}{256} x^{8} + \frac{5}{4096} x^{10} + \dots

\int e^{x} - 1 d x

\int x (x + \frac{1}{x}) d x

t an g e n t f (x) = 41 x - 16 x^{2}, at x = 1

\int \frac{1}{1 + sin ( x ) cos ( x )} d x

ma c l a u r in (1 + x)^{- \frac{6}{5}}