麦克劳林 e^{sin(x)}

解答

麦克劳林

e^{s i n (x)}

1 + x + \frac{1}{2} x^{2} - \frac{1}{8} x^{4} - \frac{1}{15} x^{5} + \dots

求解步骤

应用麦克劳林公式

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

求导数

= 1 + \frac{1}{1 !} x + \frac{1}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{- 3}{4 !} x^{4} + \frac{- 8}{5 !} x^{5} + \dots

整理后得

= 1 + x + \frac{1}{2} x^{2} - \frac{1}{8} x^{4} - \frac{1}{15} x^{5} + \dots

\int (x^{7} - 3 x^{4} + x) d x

\int \frac{1}{( y ^{2} + x ^{2} ) ^{\frac{3}{2}}} d x

\int 7 cos^{5} (x) sin^{4} (x) d x

x \to a lim (3)

d er i v a t i v e y = ln (ln (5 x))