泰勒 cos(pix),1

解答

泰勒

cos (π x), 1

- 1 + \frac{π ^{2}}{2} (x - 1)^{2} - \frac{π ^{4}}{24} (x - 1)^{4} + \frac{π ^{6}}{720} (x - 1)^{6} - \frac{π ^{8}}{40320} (x - 1)^{8} + \dots

求解步骤

应用泰勒公式

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

求导数

= - 1 + \frac{0}{1 !} (x - 1) + \frac{π ^{2}}{2 !} (x - 1)^{2} + \frac{0}{3 !} (x - 1)^{3} + \frac{- π ^{4}}{4 !} (x - 1)^{4} + \frac{0}{5 !} (x - 1)^{5} + \frac{π ^{6}}{6 !} (x - 1)^{6} + \frac{0}{7 !} (x - 1)^{7} + \frac{- π ^{8}}{8 !} (x - 1)^{8} + \dots

整理后得

= - 1 + \frac{π ^{2}}{2} (x - 1)^{2} - \frac{π ^{4}}{24} (x - 1)^{4} + \frac{π ^{6}}{720} (x - 1)^{6} - \frac{π ^{8}}{40320} (x - 1)^{8} + \dots

\int_{0}^{1} \frac{x ^{4} + 1}{x} d x

\int 1 + sinh^{2} (x) d x

\int 3 cos (x) - 4 sin (x) d x

d er i v a t i v e 6 e^{6 x}

a re a y = 5 - x^{2}, y = x^{2} - 3