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given the definite integral from negative 1 to 3 of f of X DX is equal to negative 2 and the definite integral from negative 1 to 3 of G of X DX is equal to 5 what is the definite integral from negative 1 to 3 of 3 f of X minus 2 G of X DX all right so to think about this what we could use as some of our integration properties and so the first thing that I would want to do is we could split this up into two integrals we know that the and this is true of definite or indefinite integrals that the integral of f of X 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 minus so we could split this up in the same way so this is going to be equal to the definite integral from negative 1 to 3 of 3 f of X DX minus the integral from negative 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 we can take the scalars we're multiplying the functions on the inside by these numbers 3 & 2 and we can take those outside of the integral and that comes straight out of the property that if I am 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 and so I can rewrite this as so let's see I can rewrite this first integral as 3 times the definite integral from negative 1 to 3 of f of X DX minus 2 times the definite integral from negative 1 to 3 of G of X actual let me do the second one in a different color - - this going to be the magenta - 2 times the integral from negative 1 to 3 of G of X DX and so what is this going to be equal to well they tell us they tell us what this thing is here that I'm that I'm underlying in orange the integral from negative 1 to 3 of f of X D DX they tell us that that is equal to negative 2 so that thing is negative 2 and likewise this thing right over here the definite integral from negative 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 and so the whole thing is going to be 3 times negative 2 which is equal to negative 6 minus 2 times 5 minus 10 which is equal to negative 16 and we're done