Solve Hairy Trig Integral: Find Value of 'c

[Saurabh]

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where a, b, c, d and n, all are positive integers.
Find the value of 'c'.
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I don't really have a good approach for this one.
I just made a substitution u = sinx + cosx
I couldn't clear up the mess.
A hint(s) would be highly appreciated.
Peace!

 

[Ray Vickson]

<Moderator's note: Moved from a technical forum and thus no template.>

https://scontent.fhyd1-1.fna.fbcdn....=1b8a98481cf7145ee977f8972836ac1b&oe=5A9C989E
My approach to the monster problem.
I set u as (sinθ + cosθ)
thus the limits vary from √2 to (√3 +1)/2
then i factored the sin^3(θ) - cos^3(θ) term.
and did stuff.
observe du = cosθ - sinθ = -(sinθ - cosθ)
and thus somehow the limits may change. and from that, i don't know, magically, 'c' can be equal to 3.
a hint would be appreciated :)

You really need to show your work; just saying "...did stuff..." is not enough.

Also, you should try to avoid posting images; the preferred mode here is typed problem statements and solutions, although to some extent those standards are relaxed if images are clear and unambiguous. Yours is a bit fuzzy: the lower integration limit is not at all clear.

 

[jedishrfu]

We use latex to enter equations on PF.

There is a link to our latex help page at the bottom of my post.

 

[Saurabh]

You really need to show your work; just saying "...did stuff..." is not enough.

Also, you should try to avoid posting images; the preferred mode here is typed problem statements and solutions, although to some extent those standards are relaxed if images are clear and unambiguous. Yours is a bit fuzzy: the lower integration limit is not at all clear.

Is this okay now?
Please can you figure out something?

 

[Saurabh]

@haruspex please help me with this one sir!
a little help will be appreciated.
thank you.

 

[tnich]

You might have something with your substitution. I am interested to see how you substituted for terms like ##cos^2x##. Could you show us how you did that and what you got for the integral in terms of u?

 

[haruspex]

@haruspex please help me with this one sir!
a little help will be appreciated.
thank you.

So far it has me beat. My guess is that you need to obtain a recurrence relation. That would normally be via integration by parts, but the denominator makes that tough.
Will think about it some more.

 

[tnich]

Assume the indefinite integral is ##\frac 1 n (f(x))^n##. Then the integrand must be ##f'(x)(f(x))^{n-1}##.

 

[Saurabh]

Thanks to everyone who tried.
I got the solution :)
My teacher laughed at me for not getting such a simple one.
Anyways,
the answer is c = 8.

 

[Ray Vickson]

Thanks to everyone who tried.
I got the solution :)
My teacher laughed at me for not getting such a simple one.
Anyways,
the answer is c = 8.

How did you get the answer? Many of us have tried and failed!