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  1. H

    ##\int (\sin x + 2\cos x)^3\,dx##

    Homework Statement $$\int (sinx + 2cos x)^3dx$$ Homework Equations The Attempt at a Solution $$\int (sinx + 2cos x)^3dx$$ $$\int (sinx + 2cos x)((sinx + 2cos x)^2dx)$$ $$\int (sinx + 2cos x)(1 + 3cos^2x+2sin2x)dx$$ How to do this in simpler way?
  2. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Hmm..... i see... but. Its fun anyway. Thanks for all the disscussion before. :D
  3. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    I have no idea.. I just find it randomly.. and i dont even understand this is how it work actually
  4. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    I searched on internet. Something says : if limit exist for sequences a_n. Then all the subsequences have the same limit as a_n So. Lim n->##\infty## a_n = 2/3(Lim n->##\infty## a_n ) + 1/4 So 1/3(Lim n->##\infty## a_n ) = 1/4 Lim n->##\infty## a_n = 3/4 Is it?
  5. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Can i just get the solution please? I have no idea what we discuss..
  6. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Ok. What is it? Is there any clue for me?
  7. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Ok. I can't use ##\infty## on sequence formula because it just not appropriate mathematically?
  8. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Is there any clue to get the ##a_n## formula with only n and numbers in it? $$ a_\infty = \frac{2}{3}a_{(\infty-1)}+\frac{1}{4} $$
  9. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Yes. But what do you mean by that 2 examples.. as what i explained in #19. I don't get that
  10. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    The sequence getting smaller as n approaches ##\infty## but the formula i guessed is getting the bigger value as n approaches ##\infty## is that what you meant? That is why it's wrong?
  11. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    I don't understand.. difference between ##a_{n+1}(a_{n-1})## and ##a_{n+1}(a_{n-2})## ? The difference is changing according to n. ##a_{n+1}(a_{n-1})## means, i fill a at (n-1) into the formula ##a_{n+1}## right? Then why the value of ##a_{n+1}(a_{n-1})## = ##a_{n+1}(a_{n-2})## ? Since...
  12. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    why on that expression, ##\lim_{n \to \infty} a_n = L## and NOT ##\lim_{n \to \infty} a_n = \lim_{n \to \infty} L##
  13. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Sorry. IT HAS TO BE ##\lim_{n \to \infty} ## ALL THIS TIME. NOT ##\lim_{x \to \infty} ## . i made mistake in the question
  14. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    I guess, the formula for ##a_n## will be ##(\frac{2}{3})^{(n-1)}a_1 + (n-1)\frac{1}{4}## = ##3.(\frac{2}{3})^{(n-1)} + \frac{1}{4}(n-1)## Take the limit for ##a_n## as x approaches ##\infty## i dont get what the question wants... Usually taking the limit as x approaches ##\infty## , i have to...
  15. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    ##a_n## approaches a limit ##L##. What does it mean? Lim of ##a_n## the same as limit ##L##? Or ##a_n## has value of ##L##? And take the limit as x approaches ##\infty## ?
  16. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Homework Statement ##a_1## = 3 ##a_{n+1}## = ##\frac{2}{3} a_n + \frac{1}{4} ## Homework Equations [/B] The Attempt at a Solution Sequence i got : ## a_1, a_2, a_3, a_4, a_5 ## ## 3, \frac{9}{4}, \frac{7}{4}, \frac{17}{12}, \frac{43}{36} ## I tried to find the formula of ##a_n## ##a_n =...
  17. H

    How to solve this integral of an absolute function?

    Homework Statement Homework Equations The Attempt at a Solution I think the answer for number 1 , graph somewhat like this I get trouble for 2, 3, etc I (k) = ##\int_{-1}^{1} f(x) dx ## f(x) = ## \mid x^2 - k^2 \mid## 2) k < 1 for negative side ##\int_{-1}^{-k} (x^2 - k^2) dx +...
  18. H

    Find f(x) which satisfies this integral function

    ##f(x) = x + \frac{1}{\pi} \int_{0}^{\pi} f(t) \sin^2{t} \ d(t)## ## \int_{0}^{\pi} f(t) \sin^2{t} \ d(t)## = constant I just get that it is constant because its variable doesn't depend on any x variable in f(x). because its variable is t . Is it true? set ## \int_{0}^{\pi} f(t) \sin^2{t} \...
  19. H

    Find f(x) which satisfies this integral function

    The integral of k is ##\pi^2##/4 The f(x) = t + k/##\pi## f(x) = x + ##\pi##/4 But The right answer is f(x) = x + ##\pi##/2 Set f(t) = u and d(u) = 1 ##\frac{1}{2}(1-\cos2t) ##= d(v) and v = ## \frac{1}{2} t - \frac{\sin2t}{2} . 2## ##\int u.d(v) = u.v - \int v. d(u) ## ##\int_{0}^{\pi} f(t)...
  20. H

    Find f(x) which satisfies this integral function

    I don't know exactly to integrate it... I got ##\frac{{\pi}^2}{4}## as the answer.. but I'm not sure
  21. H

    Find f(x) which satisfies this integral function

    By taking the derivative of both sides Resulting d f(t)/dt = 1 Isn't it?
  22. H

    Find f(x) which satisfies this integral function

    ##\int_{0}^{\pi} f(t) \sin^2 t \ d(t)## = ##\int_{0}^{\pi} f(t) \frac{1}{2}(1 - \cos{2t}) \ d(t)## = ## f(t) (\frac{1}{2}t - \frac{1}{2}2sin2t) |_{\pi}^{0} - \int_{0}^{\pi} \frac{1}{2}t - sin2t \ d(f(t)) \ d(t)## what i'm not sure about is the derivative of f(t) which is the integral f(t) = x...
  23. H

    Find f(x) which satisfies this integral function

    do you mean : y = mx + c y = x + ##\frac{\int_{0}^{\pi} f(t) \sin^2t \ d(t)}{\pi}##
  24. H

    Find f(x) which satisfies this integral function

    hmm, the integral contains x, still can be zero?
  25. H

    Find f(x) which satisfies this integral function

    yes, i've confirmed it yes it's ##\pi## , if it were x, then integral f(t) equals to f(x) with the integral sin.. or kind like that.. i'm still trying
  26. H

    Find f(x) which satisfies this integral function

    Homework Statement find f(x) which satisfies f(x) = x + ##\frac{1}{\pi}## ##\int_{0}^{\pi} f(t) \sin^2{t} \ d(t)## Homework Equations The Attempt at a Solution to solve f(x), I have to solve the integral which contains f(t). And f(t) is the f(x) with variable t? if yes, I will get integral...
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