Answer:
We note that,
x/³√(1+x²) dx = (3/4) d/dx (1+x²)²ᐟ³
now we can use binomial series on (1+x²)²ᐟ³
(1+x²)²ᐟ³ = ( 1+2x²/3+((2/3)*(2/3 -1)/2) x⁴ + ((2/3)*(2/3 -1)(2/3–2)/6) x⁶ +o(x⁶) =
= 1 + 2x²/3 - x⁴/9 +4x⁶/81 +o(x⁶)
The last step is to differentiate,
x/³√(1+x²) dx = (3/4) d/dx (1+x²)²ᐟ³
= (3/4) d/dx (1 + 2x²/3 - x⁴/9 +4x⁶/81 +o(x⁶) )
= (3/4) ( 0 + (4/3)x - 4/9 x³ + 24x⁵/81 + o(x⁵))
= x - x³/3 + 2x⁵/9 + o(x⁵)
The complete Question is- How do I find the Maclaurin series using binomial series in the function f(x) = x/³√1+x^2?
Learn more about binomial series here:
https://brainly.com/question/14004514
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