If f(x) is odd and ∫8−3f(x)dx=14, then
∫^8_3f(x)dx=
We can use the property that the integral of an odd function over a symmetrical interval (e.g. from -a to a) is always 0.
So,
∫^-3_8 f(x) dx = ∫^8_-3 f(-x) dx
Since f(x) is odd, we have f(-x) = -f(x)
So,
∫^-3_8 f(x) dx = -∫^8_-3 f(x) dx
And we know that ∫8−3f(x)dx = 14
So,
-∫^8_-3 f(x) dx = 14
Multiplying by -1 on both sides, we get:
∫^8_-3 f(x) dx = -14
Finally, using the property mentioned earlier, we can change the limits of integration to get:
∫^8_3 f(x) dx = -∫^-3_8 f(x) dx = -(-14) = 14
Therefore, ∫^8_3 f(x) dx = 14.