Evaluate:
1 cos x
∫ ------------
-1 e(1/x) +1
f(x)=cos(x)/[e^(1/x)+1]
The integral of f(x) from -1 to 1 is the sum of integrals from -1 to 0 and from 0 to 1.
A graph suggests we get the negative half by flipping things and integrating
f(-x)=cos(x)/[e^(-1/x)+1] from 0 to 1.
The we can sum f(x)+f(-x) and integrate this from 0 to 1.
What interesting is the denominators of f(x) and f(-x) have the property of have the same sum and product: 2+e^(1/x)+e^(-1/x). So the sum is simply
f(x)+f(-x)=cos(x)
And so the integral is sin(1)-sin(0) = sin(1 radian) = 0.841
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Posted by Jer
on 2023-12-26 14:05:39 |
Solution
