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Popular >

taylor cos(x)+1/x , pi/2

Solution

taylor

cos (x) + \frac{1}{x}, \frac{π}{2}

Solution

\frac{2}{π} + \frac{- π ^{2} - 4}{π ^{2}} (x - \frac{π}{2}) + \frac{8}{π ^{3}} (x - \frac{π}{2})^{2} + \frac{π ^{4} - 96}{6 π ^{4}} (x - \frac{π}{2})^{3} + \frac{32}{π ^{5}} (x - \frac{π}{2})^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= \frac{2}{π} + \frac{- 1 - \frac{4}{π ^{2}}}{1 !} (x - \frac{π}{2}) + \frac{\frac{16}{π ^{3}}}{2 !} (x - \frac{π}{2})^{2} + \frac{1 - \frac{96}{π ^{4}}}{3 !} (x - \frac{π}{2})^{3} + \frac{\frac{768}{π ^{5}}}{4 !} (x - \frac{π}{2})^{4} + \dots

Refine

= \frac{2}{π} + \frac{- π ^{2} - 4}{π ^{2}} (x - \frac{π}{2}) + \frac{8}{π ^{3}} (x - \frac{π}{2})^{2} + \frac{π ^{4} - 96}{6 π ^{4}} (x - \frac{π}{2})^{3} + \frac{32}{π ^{5}} (x - \frac{π}{2})^{4} + \dots

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f (x) = x^{2} + 18 x + 86, at x = 0

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\int - \frac{1}{2} e^{- 2 x} d x

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\int_{0}^{x} \frac{1}{1 + t ^{2}} d t

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\int \frac{- x ^{3} - x ^{2} - 4 x + 28}{x ^{3} + 4 x} d x

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