00:01
So for this problem we need to solve this specific definite integral without using integration by parts.
00:09
Okay, the integron is x squared minus 1, old and square root divided by x cubed.
00:18
Okay, the trigonometric substitution i would like to suggest is the following one and i will explain why.
00:28
Let's let's suppose x equals 1 over cyan theta.
00:40
Okay, one may recognize that, okay, this is the definition of the cost again.
00:52
But in any case, i'm going to use it in this form 1 over cyan theta.
00:57
And the justification is that we need a certain expression, we offer trigonometric, a certain trachonometric expression that will that can provide, yeah, can go up to infinity.
01:13
Okay, and if we use, i don't know, sine theta or cosine theta, yeah, this go up to, from minus one to up to one.
01:23
They cannot go up to infinity.
01:26
So, yeah, this infinity there is one indication that we might better use an expression.
01:33
Like this okay so the first thing we do the first thing we do is yeah the substitution x squared so that will give one over sine square minus one square root and actually before let's work out this expression explicitly let's change colors so this one can be written as 1 minus sine theta squared over sine theta squared square root, okay and the whole expression i hope it is clear that it is cosyl theta absolute value sine theta absolute value okay then let's do another thing let's calculate dx and d -thea so dx dx equals yeah minus so on if one yeah there's a calculation sine theta squared and the derivative of sine theta with vector theta score sine theta there there okay and this way will have calculated an expression for d x mm -hmm and the last thing we need to do before the replacement actually is that we check with the integration limits.
03:34
So which value of thera, which value of thera can give minus 1? okay, x equals 1, sign, thera.
03:51
So we need a value of sine thera, for which, we need the value of thera for which, that will be minus 1.
03:59
And this of course, this is for thera.
04:03
Equals to 3 times pi over 2 or this corresponds also to minus pi over 2 so let's let's keep it this way i hope this is clear and what might be yeah and a value for theta okay apologies this is this is yeah i mean there i'm i'm there i'm yeah minus 1 i'm i'm assuming that we're keeping it inside the range of let's say theta to 2 pi okay mm -hmm and also so yeah and for this reason the the value that can i can give us x equals minus affinity it is actually so this might be 2 pi and let's use the notation minus indicating that this is the limit this is the limit from from the left approaching approaching theta it goes to pi from the left another way to write it is my zero and again we'll minus indicating that we're approaching it from the left.
05:50
If we do this, okay, sine theta becomes zero, one over zero becomes infinity.
05:56
But because we are approaching it from the left, sine theta has negative values and while it diverges while it goes to infinity, yeah, one over one over zero, excuse me, when a sine theta goes to zero and but having negative values and while convergent to zero the once all of a sin theta expression diverges but to diverges to minus infinity.
06:31
Okay, let's use all this now so limits the limits the corresponding limits are minus 0 and minus pi over 2 then coshen 30 absolutely value 1 over 7 0 0 0 cubed and d x minus 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 okay, let's do a few simplifications.
07:37
So with this one, yeah.
07:40
And now, in this specific interval for theta from minus power 2 up to 0 there.
08:02
Yeah, this is the fourth quadrant and the cosine there is positive.
08:07
Positive.
08:10
Okay, so we can actually exclude, we can actually ignore the absolute value signs there, but sign theta it is negative.
08:21
So we need to keep, yeah, we need the same theta.
08:31
Either we're going to keep the absolute values or we will need to write it as minus sign theta.
08:37
But before that, since we have this minus one there, we can free flip, we can flip the integration limits as minus pi where two, up to zero.
08:52
Then, okay, let's write, you know, let me write it, sign theta, sign, theta...