f(x) is a even function. Is it true that :
Integrating f(x) from x = a to x = b is equal to integrating f(x) from x = - b to x = - a? Please prove.
Integration
2012-07-04 1:51 pm
回答 (2)
2012-07-04 3:56 pm
✔ 最佳答案
by substitute u = -x, du = -dx,when x = a, u = -a; when x = b, u = -b
thus, integrating (from x = a to x = b) f(x) dx
= integrating (from u = -a to u = -b) f(-u) (-du)
= integrating (from u = -b to u = -a) f(-u) du
= integrating (from x = -b to x = -a) f(-x) dx
= integrating (from x = -b to x = -a) f(x) dx, since f(-x) = f(x) when even
So, it is true
參考: knowledge
2012-07-05 3:54 am
Let y = -x
∫ (a -> b) f(x) dx
= - ∫ (-a -> -b) f(-y) dy
= - ∫ (-a -> -b) f(y) dy (as f(x) is a even function)
= ∫ (-b -> -a) f(y) dy
= ∫ (-b -> -a) f(x) dx (as y is a dummy variable)
∫ (a -> b) f(x) dx
= - ∫ (-a -> -b) f(-y) dy
= - ∫ (-a -> -b) f(y) dy (as f(x) is a even function)
= ∫ (-b -> -a) f(y) dy
= ∫ (-b -> -a) f(x) dx (as y is a dummy variable)
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