Final answer:
To evaluate the indefinite integral as a power series, use the Maclaurin series expansion of the function and integrate term by term. The radius of convergence for the power series is |x| < 1.
Step-by-step explanation:
To evaluate the indefinite integral as a power series, we can use the Maclaurin series expansion. The Maclaurin series expansion of tan⁻¹(x) is given by:
tan⁻¹(x) = x - x³/3 + x⁵/5 - x⁷/7 + ...
Integrating term by term, we get:
f(x) = ∫ tan⁻¹(x⁴) dx = ∫ (x⁴ - x¹²/3 + x²⁰/5 - x²⁸/7 + ...) dx
Simplifying this expression, we have:
f(x) = x⁵/5 - x¹³/39 + x²¹/105 - x²⁹/203 + ...
The radius of convergence, r, for this power series is determined by finding the interval of x-values for which the series converges. In this case, the series converges for |x| < 1.