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CalculusCh8.2 Integration By PartsName:Let's start with the product rule for derivatives:d[()g(x)] = f "(r)g(r)+ f(r)g 'x)Integrate both sides: f()g(...

Question

CalculusCh8.2 Integration By PartsName:Let's start with the product rule for derivatives:d[()g(x)] = f "(r)g(r)+ f(r)g 'x)Integrate both sides: f()g(x) = f f "(x)g(xJdx+ f f (x)g "(xJdtx Rearrange terms: Jf()g'(x)dx = f()g(r)-fgtx) f "(x)trFormula for integration by parts:f(x) g(x) dv g (x)dx du = f'(xJdxudv= uVvduVideo Examples: #1,#3,#5In X dx =85Inx dx =(3) e" cosx dx=(4) 1 sin * dx =

Calculus Ch8.2 Integration By Parts Name: Let's start with the product rule for derivatives: d[()g(x)] = f "(r)g(r)+ f(r)g 'x) Integrate both sides: f()g(x) = f f "(x)g(xJdx+ f f (x)g "(xJdtx Rearrange terms: Jf()g'(x)dx = f()g(r)-fgtx) f "(x)tr Formula for integration by parts: f(x) g(x) dv g (x)dx du = f'(xJdx udv= uV vdu Video Examples: #1,#3,#5 In X dx = 85Inx dx = (3) e" cosx dx= (4) 1 sin * dx =



Answers

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
$$g(x)=\int_{2 x}^{3} \frac{u^{2}-1}{u^{2}+1} d u$$
$\left[\text { Hint } \int_{2 x}^{3} f(u) d u=\int_{2 x}^{0} f(u) d u+\int_{0}^{3 x} f(u) d u\right]$

Okay. Use the chain rule to find the driver of which is our goal. So what we end up with his G Prime of axe is equivalent to 27 X squared, minus three over nine X squared plus one minus eight X squared, minus two over four X squared, plus born.

Were given our function. F A bex is one over X and our function G of X is equal to e to the power of forex. So let's first find are derivative the back. So that's a kind of X. So we can rewrite whatever x as x the power of negative ones. So now the derivative that using the power rule is going to be negative one times x, the power of negative tooth. And we can rewrite this as negative one over X squared. Next, we're asked to find the indefinite integral of G of X. So the anti derivative of each of the four X is going to be e for the four X, but we must have coefficient of 1/4 in the front to cancel out with the four when we take a derivative. And because this is an indefinite integral, we need to add our constant C

Were given the function. F FX is equal to eat the power of forex and another function, G FX, which is equal to what other ex. So we're asked to find first a derivative of that and then the indefinite integral of G f X Lisa. Let's first start with a kind of like so the derivative of F is going to be B to the four X and then we have to change all the four x So this is going to be times for so if climate taxes for the to the four X now the indefinite integral of G of X is going to be equal to lawn of actual value of X because the anti derivative of one over X is one of the actual value of X. But because this is an indefinite integral, we have to remember to add our constancy.

All right. So we are being asked girl what for? Is knowing what F. Of one is and knowing that our derivative of F. Is X squared. So what that means is they want me to set up an integral. That looks like problem is there could possibly be a constant that I don't know about what I go to do the derivative, but this is what I've got right. So if I think about this X squared, going backwards for sure is huge. One third. Excuse. That's what I got. But when I check that one third Times 1 to the power of three right now I'm sitting at one third. But my answer has to be three. That means I have a constant of some sort so that when I do the derivative of that just cancels out anyway. So I have to kind of work backwards here to figure out what my constant was To be able to figure out what my f. of four. So what am I missing on 1/3 To get to three. And the answer for that. Cause I just have to subtract 1/3 from three. So my final derivative you could use it as two and 2/3 or could use it as 8/3 is my comment. So once we know this important piece of information, Finding F of four is really easy because that means my derivative equation is one third The power of three less 8 30. So just plugging in four, We end up with 1/3 for cube plus 8 30. So four cubed is 64. So I end up with 64/3. Let's eat three. That works out to 72/3, which does Bye 2 24 FF four would be the value 24.


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