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

#Integration #IITJEE #Integrals

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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 - 3 years, 4 months ago

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 - 3 years, 4 months ago

What makes you think that it has a closed form?

Pi Han Goh - 3 years, 4 months ago

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 - 3 years, 4 months ago

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 - 3 years, 4 months ago

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

Write \(\cos^5(x)\) and \(\cos^4(x)\) as \(\frac{1}{16}(10 \cos(x) + 5 \cos(3x) + \cos(5x))\) and \(\frac{1}{8}(3 + 4 \cos(2x) + \cos(4x))\).
Substitute \(z = {e}^{ix}\).
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 - 3 years, 4 months ago

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

Pi Han Goh - 3 years, 4 months ago

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 - 3 years, 4 months ago

@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 - 3 years, 4 months ago

@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 - 3 years, 4 months ago

@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 - 3 years, 4 months ago

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 - 3 years, 2 months ago