Let f ( x ) and g ( x ) be the functions graphed below.
f ( 0 ) ⋅ g (0) = 2 ⋅ 0 = 0.
Let x be any value less than 1. Then g ( x ) = 0, so f ( x ) ⋅ g ( x ) = 0 also.
Now let x be any value between 1 and 2. Then f ( x ) = 0, so f ( x ) ⋅ g ( x ) = 0.
The function f ( x ) ⋅ g ( x ) is the zero function.
The integral of the zero function is zero, so
No. We saw that
The integral of the product is not the product of the integrals.
The properties having to do with inequalities will get used more after we get to improper integrals. For now, we'll focus on the other properties we've seen so far.
Assume f and g are integrable functions.
We can pull out the constants:
Finally, substitute in the values of the integrals and to get
Since f is even, we know that
Since g is odd, we know that
Putting these values in where we left off gives us
with a = -3, b = 1, and c = 2, we can combine the first two terms of this expression:
Using the same property again, this time with a = -3, b = 2, and c = 3, we can combine the remaining two terms:
Since we're assuming f is odd, this integral is equal to 0.
Pull out the constant:
Switch the limits and switch the sign:
Substitute to get
Make it rain.