What is taylor arctan(x^2) ?

The answer to taylor arctan(x^2) is x^2-1/3 x^6+1/5 x^{10}+\ldots

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taylor arctan(x^2)

taylor

arctan (x^{2})

x^{2} - \frac{1}{3} x^{6} + \frac{1}{5} x^{10} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{0}{1 !} x + \frac{2}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 240}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{0}{9 !} x^{9} + \frac{725760}{10 !} x^{10} + \dots

Refine

= x^{2} - \frac{1}{3} x^{6} + \frac{1}{5} x^{10} + \dots

\frac{d y}{d x} - \frac{3 x ^{2}}{2 y} = 0

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

\int (x^{2} + 3)^{4} 2 x d x

\int e^{- a x^{2}} d x

What is taylor arctan(x^2) ?
The answer to taylor arctan(x^2) is x^2-1/3 x^6+1/5 x^{10}+\ldots