Expert: Abe Mantell - 5/27/2009

Question
Use power series to approximate the following integral to six decimal places
.5
∫ln(1+x^4)dx
0
I got end answer of .00625-.000108507   Can you tell me if this is correct and if I should now subtract it out?
Thanks


Answer
Since the power series expansion of ln(1+x) is
x-1/2 x^2+1/3 x^3-1/4 x^4+1/5 x^5-1/6 x^6+...
we can easily get the expansion for ln(1+x^4) by replacing
"x" with "x^4" to yield:
x^4-1/2 x^8+1/3 x^12-1/4 x^16+1/5 x^20-1/6 x^24+...
Thus,
int(x^4-(1/2)*x^8+(1/3)*x^12-(1/4)*x^16, x = 0 .. .5)
is about 0.0061445 (to 7 decimal places)

The true value is about 0.0061445152179695
So the value obtained above is certainly good to the desired number
of places.

Abe