Use the power series to approximate the definite integral to six decimal places.

Question

(0 to 0.3)∫ (x²)/(1+x⁴)dx

Answer 1

Tan^-1x = x - x³/3 + x⁵/5 - x⁷/7 + ...

(Tan^-1x)' = 1 / (1+x²) = 1 - x² + x⁴ - x⁶ + ...

1 / (1 + x⁴) = 1 - x⁴ + x⁸ - x¹² + ...

x² / (1 + x⁴) = x² - x⁶ + x¹⁰ - x¹⁴ + ...

So, ∫_{(0 to 0.3)} [x² / (1 + x⁴)]dx = (1/3)(0.3)³ - (1/7)(0.3)⁷ + (1/11)(0.3)¹¹ - ...