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\[\huge\int \dfrac{\cos^5 x+\cos^4 x}{1-2\cos 3x}\]

Note by Rohit Ner
2 years, 1 month ago

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I am getting a nice closed form if the numerator is \( \cos 5x + \cos 4x \). I think if it is a JEE problem then the numerator should be what I have written. Can you please check? Otherwise I'll try with this again.

Sudeep Salgia - 2 years, 1 month ago

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I am sure about the question. The reason I put the jee tag was to get to know if there are any methods of jee applicable.

Rohit Ner - 2 years, 1 month ago

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I don't think this is an IITJEE question. This question is not integrable to our knowledge(atleast till JEE point of view). You can use higher level integration to solve this.

Harikrishna Nair - 1 year, 11 months ago

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I know a method but haven't done it yet.

  1. Write \(\cos^5(x)\) and \(\cos^4(x)\) as \(\frac{1}{16}(10 \cos(x) + 5 \cos(3x) + \cos(5x))\) and \(\frac{1}{8}(3 + 4 \cos(2x) + \cos(4x))\).

  2. Substitute \(z = {e}^{ix}\).

  3. Then, you would get a rational polynomial function in terms of z which can "easily" be solved using Partial Fraction or Division approach.

I know this method is way too tedious but that's the most general way to tackle these types of problems.

Kartik Sharma - 2 years, 1 month ago

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You can't solve it by simple Partial Fractions because the denominator of this function does not have any "nice" roots.

Pi Han Goh - 2 years, 1 month ago

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Oh! I see. I hadn't checked it so I might be wrong. If it doesn't have "nice" roots then also it doesn't matter, computer will do it, it doesn't discriminate b/w real and complex :P

Kartik Sharma - 2 years, 1 month ago

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@Kartik Sharma We might have different opinion of "nice" closed form. Does the integration of \( \frac1{x^3 + 3x^2 + 5x+7} \) have a nice form?

Pi Han Goh - 2 years, 1 month ago

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@Pi Han Goh That's quite nice! :P For me, everything has a nice "closed"(I didn't use this word originally) form. Even the error function does! :P

Kartik Sharma - 2 years, 1 month ago

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@Kartik Sharma To me, even the error function is not "nice". I only consider elementary functions to be "nice".

If you consider all of these to have a "nice" closed form, then it's hard to judge whether an integral is worth solving or not. Don't you think so?

Pi Han Goh - 2 years, 1 month ago

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What makes you think that it has a closed form?

Pi Han Goh - 2 years, 1 month ago

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This problem was given to me by my class mate. I tried every integration technique possible. Even I doubt if there exists a valid closed form for it. Please help me sir.

Rohit Ner - 2 years, 1 month ago

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There's no simple closed form without using hypergeometric functions. Apply \( \cos(3x) /\cos(2x) = 1 - 2\cos(2x) \) and reducing the powers of trigonometric functions to 1 shows that we are essentially solving for at least one of \( \int \sin(ax) \csc(bx) dx \) , \( \int \sin(ax) \sec(bx) dx \), \(\int \cos(ax) \sec(bx) dx \), \( \int \cos(ax) \csc(bx) dx \) which can't be stated in terms of elementary functions because for all of these cases, \(b \ne 1 \).

Pi Han Goh - 2 years, 1 month ago

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