Question
Letf(x) dx = 5, J6 " fo)dx = 6, 6 g(x) dx =- -9, 6 g(x) dx = 3, Use these values to evaluate the given definite integrals_ J" <o)+g6)) dx " <o) -gtx)) dx K" of6) +2g6)) dx =Find the valua such that(af(x) + g(x)) dx =
Let f(x) dx = 5, J6 " fo)dx = 6, 6 g(x) dx =- -9, 6 g(x) dx = 3, Use these values to evaluate the given definite integrals_ J" <o)+g6)) dx " <o) -gtx)) dx K" of6) +2g6)) dx = Find the valua such that (af(x) + g(x)) dx =
Answers
Evaluate the integrals. $$ \int_{6}^{9}\left(5-x^{2}\right) d x $$
So for this problem in order to evaluate are fairly large integral here. It's going to be easier if we break this part into a separate into girls. So using our properties of intervals, this is equal to on the interval of five extra fourth RDX minus the integral of three X squared DX plus the in a roll of two x t x, my lips not to x X and two ex d x was the integral of 60 x. And so from here, we're going to be able to use our properties of inter girls in our power rule to simplify. So the integral of five X to the fourth power. You know, we could pull out that constant five. I'm so it's just the interval of X, the fourth power. So using the power rule, this is going to be won over four plus one, which is five. I'm excited for plus one power, which is again five um, minus three. It will pull out this constant again times I used the powerful one over two plus three of two plus one. She's 1/3 times x to that same three power um, plus the interval of two x so plus X squared and finally plus interval of six will be plus six X and then lastly, we can't forget our constant of integration seat. And so when we distribute out or must play out all these constants that we can see that they're all going to go with So the integral of our function is X the fifth power minus X cute plus X squared plus six x plus c.
We're going to use partial fractions. Will have a tendency Times X plus two. Yeah, it looks good. So there's a building near factors. We're gonna have a on Rex minus C plus B over X plus two on the equations that resides to law. Here's the situation. Any times exposed to be times X man thing end up by eight plus B should equal high and also two and minus A B equals zero. So we cannot supply the situation by three and add it to the other to eliminate the bottom. So we end up with that's a 15 on the right hand side to be cancelled. Yeah, five it. So we have that a is equal, Teoh. Fine. But if is equal physical ability What a crazy close. The three inlets. Like an equation being my sweetie bulls to. So we have that. This animal is the same, uh, taking the interval of dream over X ministry. Plus, what's it to over at lusted the X. We take this, we're gonna get the times the natural log, Lex minus three plus two times and that you're alive. Uh, exposed to now. We could We wanted Teoh bring them together. They have bringing the coalitions and its power, But this looks good
So we're given these two for F and fruity To find from 05 for both. And so first part is asking us to find 0 to 5 of half plus G. So it's just going to be equal to six plus two which equals eight. Okay, Next one. So asking for the same this time with F minus G. And so then we just evaluate accordingly. That's going to be six minutes to which was the floor next part. So we've got for C negative four times And a girl from 0 to 5 of F. We can pull out the constant since that's part of the integral properties here. And so That's going to be a -4. Type six Equals 10: 24. Okay, last one. So from here we're finding integral from 0 to 5 of half minus three times G. Again we can pull out that constant and then evaluate F and G Per what we have. So we have 6 -2 times three Which equals or three times 2 Basically is 6 26 which is zero. So those are answers and return
Hello Friends. We have to find the definite in legal. That is modified 21 six and two. You squirreled off six months 2 x. dx. Okay so now we will calculate it -5-1 Square root of six months 2 weeks dx. So this will be close to six interrogation of this will be six months to X. to the power of one way to plus one upon one way to plus one. Okay. Into you know into we went up one managed to Okay And the limit is -5 grand one so we can solve it six months 2 weeks to the power of three were to upon Mhm 32 Dumanis too -5: one. So this will be close to six six months 2 weeks to the bar. Aww three where they were born, monastery. Okay so now we will put the limit So this will be minus of two six months to the power off 3 2 months off six minus two in 25. That is 10 To the power of three x 2. So this will request two months of 26 months to that is full for to the bubble three x two months of 16 to the power of three x 2. Minister. 4 to the power three vital will be yet 16 2. Power three word will be 64 64 -8 Will be 56 Into two. That will be costume 112 So this is the and that I hope you understood. Thank you.