Definite integral properties (no graph): function combination | AP Calculus AB | Khan Academy

Given that the definite integral from -1 to 3 of f of x dx is equal to -2, and the definite integral from -1 to 3 of G of x dx is equal to 5, what is the definite integral from -1 to 3 of 3 f of x - 2 G of x dx?

All right, so to think about this, what we could use is some of our integration properties.

The first thing that I would want to do is split this up into two integrals. We know that, and this is true of definite or indefinite integrals, that the integral of f of x plus or minus G of x dx is going to be equal to the integral of f of x dx plus or minus the integral of G of x dx. If this is a plus, this is going to be a plus; if this is a minus, this is going to be a minus.

So we could split this up the same way. This is going to be equal to the definite integral from -1 to 3 of 3 f of x dx minus the integral from -1 to 3 of 2 G of x dx. Notice all I did is I split it up. Taking the integral of the difference of these functions is the same thing as taking the difference of the integrals of those functions.

Now, the next thing we can do is take the scalars we're multiplying the functions on the inside by these numbers, three and two, and we can take those outside of the integral. That comes straight out of the property that if I'm taking the integral of some constant times f of x dx, that is equal to the constant times the integral of f of x dx.

So I can rewrite this as—let's see—I can rewrite this first integral as 3 times the definite integral from -1 to 3 of f of x dx, plus 2 times the definite integral from -1 to 3 of G of x dx. Actually, let me do the second one in a different color—minus, this is going to be magenta—minus 2 times the integral from -1 to 3 of G of x dx.

So what is this going to be equal to? Well, they tell us what this thing is here that I'm underlying in orange: the integral from -1 to 3 of f of x dx is equal to -2, so that thing is -2. Likewise, this thing right over here, the definite integral from -1 to 3 of G of x dx, they give it right over here; it's equal to 5, so that's equal to 5.

Therefore, the whole thing is going to be 3 times -2, which is equal to -6, minus 2 times 5, which is -10, and that's equal to -16. And we're done.