Use a power series to approximate the definite integral to s

Use a power series to approximate the definite integral to six decimal places. The integral from 0 to 0.4 of ln(1+x^4)dx

Solution

the geometric series 1/(1 - t) = S(n=0 to 8) t^n. Replace t with (-t): 1/(1 + t) = S(n=0 to 8) (-1)^n t^n. Integrate both sides from 0 to t: ln(1 + t) = S(n=0 to 8) (-1)^n t^(n+1)/(n+1). Reindex the sum and let t = x^4: ln(1 + x^4) = S(n=1 to 8) (-1)^(n-1) x^(4n)/n. Therefore, ?(0 to 0.4) ln(1 + x^4) dx = S(n=1 to 8) (-1)^(n-1) (0.4)^(4n+1)/[n(4n+1)].