Finding power series and integral value

Question

Find the power series of ln(x^(2) + 1), and with that estimate the value of ∫ln(x^(2) + 1)dx (with an upper bound of 0.5 and lower bound of 0)

Answer 1

Taylor Series of ln(1+x) = x - x²/2 + x³/3 - ... (Derived from def'n of Maclaurin Series)

Taylor Series of ln(1+x²) = x²- x⁴/2 + x⁶/3 - ... (By substitution of x² for x)

You can find the general integral of the TS by integrating each term

Integral of TS of ln(1+x²) = x³/3 - x⁵/10 + x⁷/21 - ... = Iapprox

Now just evaluate Iapprox from 0 to .5 i.e. Iapprox(.5) - Iapprox(0)

You can use the number of terms that get you close enough... (The numerical integral is .038867 for comparison)

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