What is taylor xcos(5x) ?

The answer to taylor xcos(5x) is x-25/2 x^3+625/24 x^5-3125/144 x^7+78125/8064 x^9+\ldots

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taylor xcos(5x)

taylor

x cos (5 x)

x - \frac{25}{2} x^{3} + \frac{625}{24} x^{5} - \frac{3125}{144} x^{7} + \frac{78125}{8064} x^{9} + \dots

Solution steps

Apply the Taylor Formula

= 0 + \frac{\frac{d}{d x} ( x cos ( 5 x ) ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( x cos ( 5 x ) ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x cos ( 5 x ) ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 75}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{3125}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- 109375}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{3515625}{9 !} x^{9} + \dots

Refine

= x - \frac{25}{2} x^{3} + \frac{625}{24} x^{5} - \frac{3125}{144} x^{7} + \frac{78125}{8064} x^{9} + \dots

(y + x + \frac{y ^{2}}{x}) d x - x d y = 0

\int 6 sin (x) + 5 d x

\int x 2^{- x} d x

\frac{d y}{d x} = 3 x^{- 4}

\frac{d y}{d x} = 2 y - 3 y^{2}

What is taylor xcos(5x) ?
The answer to taylor xcos(5x) is x-25/2 x^3+625/24 x^5-3125/144 x^7+78125/8064 x^9+\ldots