00:01
All right, in this problem, we're going to take look at this improper integral.
00:04
We know it's improper because of the upper limit is infinity.
00:08
And we're going to use a technique to find the antiderivative first through partial fractions.
00:16
And so we're just going to look at this integral, and we're going to take the fraction 1 over x times 2x plus 5, and we'll split it into a over x plus b over 2 .8.
00:31
X plus five and then we will multiply everything by x times 2x plus five and and then simplify.
00:45
So what we get is 1 equal to a multiplied by 2x plus 5 and we're going to add b times x.
00:55
And we can let x be anything here.
00:58
We can make life a little easier by choosing.
01:02
X to be zero and that will cancel b and so what we would get there is 1 equals a multiplied by 5 so a is going to be 1 5th and then we will also do the same kind of thing except this time we'll let x equal uh let's see negative uh 5 halves and that would cancel a and that would give us 1 equals b times negative 5 halves so b is equal to negative two -fifths.
01:42
All right, so what this means then is that the original integral that we had could be then written as a, which is one -fifth, so we'll do one over five -x minus b, so b is two over five, and that's divided by two x plus five.
02:05
And so we have a slightly reduced integral that is a little easier to deal with.
02:11
Okay, so the anti -derivative of this one now is 1 -5th, the natural log of the absolute value of x minus 2 -fifths times the natural log of the absolute value of 2x plus 5, and then we will have to multiply this by 1 -half to take account for that 2x in there.
02:35
So in the anti -derivative piece of this problem, we have one -fifth l -n at absolute value of x minus one -fifth, the natural log of 2x plus 5.
02:51
And we can certainly simplify this, and it very well may help us.
02:55
So let's go ahead and factor out the 1 -5th.
02:58
And then we'll take the natural log, and with the subtraction, we're going to divide the arguments...