Representations of functions as power series

Question

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

∫ (from 0 to 0.3) x⁶ / (1+ x⁷) dx

i = ?

Answer 1

Power series for x⁶/(1+x⁷) = x⁶∑^inf_n=0(-1)ⁿ(x⁷)ⁿ ≈ x⁶-x¹³+x²⁰ -+.... = g(x)

∫^0.3₀g(x) dx = x⁷/7-x¹⁴/14+x²¹_/21|^0.3₀≈(0.3)⁷/7-(0.3)¹⁴/14+(0.3)²¹/21-+....

Approximately (0.3)⁷/7 ≈ 0.0000312