Question
Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.$$int x^{3} sin x d x$$
Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers. $$ int x^{3} sin x d x $$
Answers
Use any method to evaluate the integrals.
$\int x \cos ^{3} x d x$
To use a computer algebra system. Two. Evaluate this. Improper, Integral. So before we go to the computer, let's just think about why is this improper? Well, we don't have infinite limits, but we do have a singularity when X is equal to zero because we're dividing one over X. However, that's not really his bigger problem, as it might seem, because if we take the limit of this thing as X goes to zero by the Squeeze Theorem, for instance, no sign of anything is always gonna be bounded between one and minus one. And so, by the squeeze there, um, I mean that for positive values of X, this is between X and Naina sex, and both of these are approaching zero. So that means that this has to approach zero also. Okay, so we should expect that this is giving us a finite area. And so this improper integral should converge. So now let's just go to Mathematica to confirm that maybe the first thing would be just a plot dysfunction it came from. I'm from Orem argument that the limit approaches zero as X approaches zero. So let's plot this between zero and two over. Pi okay. And as you can see, it really is oscillating and even more so. It's also confirm, or estimates that it's between X and minus X if we plot these two straight lines. In addition, we do see that it is always following between. So basically, the sign part is making it obsolete. But it's forced. It's oscillation can only go between the orange and green lines and those are both going to zero. So the blue curve is forced to go to zero also. So now maybe let's just get on approximation for the integrate one over X exes from 0 to 2 over pi. And so we see we get about 0.1 Is the area between these two? Uh, sorry is the definition immoral of X Sign of one over X.
So this question we're gonna use our table of chemicals and the formula Co sign Cute co sign Human of you Year is gonna be equal toe 1/3 times two plus coastline spirit of you 1/3 to co sign Yeah, me too. To Moscow Centre part of you to cross the sign Scared of you And then this part is multiplied by sine you And then we add our finally our constant C constancy So we can do here is we can let you equal, Axl. It's right that over here, using people acts so important we can on basically to simplify that this down to 2/3 sign X plus 1/3. So the 2/3 sign axe Did you stop to be sent by 1/3 times 2 2/3 on to third Sign X plus 1/3 co sign squared at sign X co sign squared X science. Um, yes, 02 2/3. 1/3 Coast expert Jack Sign X plus our constancy. And just like the stuff that we kind of didn't show here was that we did 1/3 time's too, which is about 2/3 here and then 1/3 co sign squared Axe Everybody made you be more action than me. The reason we multiply both these by sign access because of this right here. This is our final answer.
So the question we're asked to solve the cynical using our kind for there were getting cast. So we're gonna planning on enrolling Tara back here, which will be ax squared. This were s word minus nine over three ox. Yes. And then when we put this into our chemical later, the first thing to get us, we get spirit of expert by Stein over three. On an addition to this, we have the square root of actually my sign over three minus in firsthand expert Afghanistan over three minus in first hand, X rayed minus nine. However, there is also this very weird, Einstein, and finally crossed our constant c. So this is our answer usually cast without you times officer.
In this question. We're looking for the indefinite integral of one over X. It exploded 1/2 sex to the exponents 1 30 So and this question will use a computer algebra system or and online calculator system. Wolfram Alpha is a good one to use. So making sure we have exponents and brackets so that we have the whole fraction as an exponents on the X and bracket around the whole denominator. Wait, See a dancer and we're looking for is given here, and that's what we need to do. Also note that we can use integration tables to find the answer, except we may have to multiply by some exponents of X, multiply the top and bottom of the fraction by some exponents of X to get it in the form written and the table