00:01
In this problem we are given the following integral to compute.
00:07
Integral dx 3 times exponential 2x divided by exponential 2x plus 9 exponential x plus 18.
00:26
We are asked to make a substitution to express this integrant as a rational function.
00:33
And only after this substitution we will evaluate the integral.
00:39
So let's try to do that.
00:45
First let me put this three -factor in front because it looks a bit annoying in the integrant like this.
01:02
So we notice something here.
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We have some term exponential 2x that looks like the square of this term exponential x.
01:13
So i want to define some u equal to exponential x so that i will have u squared plus 9 u plus 18 here in the denominator so that it will at least partial look like this integrant will partially look like a rational function so we will see what we are going to get in the numerator accordingly so this is what we have we have you equal to exponential x d u equal to exponential x d x or u times d x so d x is equal to d u over u okay this interval becomes three times d u over u okay we have u squared divided by u squared plus 9 u plus 18 we have some nice simplification over here and let us try to write down the denominator in a simplified form so we need two numbers whose product will give us 18 and whose sum will give us minus and not minus plus 9 so we will you can go for u plus 3 times u plus 6 to simplify the denominator in the following manner now let us right down, this simplified form clearly.
03:07
U over u plus 3 times u plus 6.
03:11
So we have this rational function.
03:16
Then the trick is to perform an expansion over this partial fractions.
03:26
So this is one method to do that...