麦克劳林 xe^{x^2}

解答

麦克劳林

x e^{x^{2}}

x + x^{3} + \frac{1}{2} x^{5} + \frac{1}{6} x^{7} + \frac{1}{24} x^{9} + \dots

求解步骤

应用麦克劳林公式

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

求导数

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{6}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{60}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{840}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{15120}{9 !} x^{9} + \dots

整理后得

= x + x^{3} + \frac{1}{2} x^{5} + \frac{1}{6} x^{7} + \frac{1}{24} x^{9} + \dots

\int_{1}^{x} sec (t) d t

\frac{d}{d x} (\frac{10}{27 x ^{\frac{8}{3}}})

t an g e n t f (x) = (1 + 4 x)^{9}, (0, 1)

d er i v a t i v e f (x) = - 3 (2 x - 2) = 0

\frac{d}{d x} (\frac{x ^{4}}{4} - 2 x^{2})