麦克劳林 e^xcos(x)

解答

麦克劳林

e^{x} cos (x)

1 + x - \frac{1}{3} x^{3} - \frac{1}{6} x^{4} - \frac{1}{30} x^{5} + \dots

求解步骤

应用麦克劳林公式

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

求导数

= 1 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 2}{3 !} x^{3} + \frac{- 4}{4 !} x^{4} + \frac{- 4}{5 !} x^{5} + \dots

整理后得

= 1 + x - \frac{1}{3} x^{3} - \frac{1}{6} x^{4} - \frac{1}{30} x^{5} + \dots

\frac{d}{d t} (2)

d er i v a t i v e f (x) = (- 9 x^{3} + 2 x^{2})^{5}

\int e^{x} \cdot 2 d x

d er i v a t i v e \frac{x}{x + 1}

d er i v a t i v e (\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) (y - 3 y^{3})