Trigonometric integral question

[tor116]

I'm trying to find the indefinite integral of cot[sup:1uuimzq2]3[/sup:1uuimzq2]x dx. I make the substitution of cot[sup:1uuimzq2]2[/sup:1uuimzq2]x = (csc[sup:1uuimzq2]2[/sup:1uuimzq2]x -1) and after multiplying it out I end up with 2 integrals. I have a question about one of them, the indefinite integral of cot x csc[sup:1uuimzq2]2[/sup:1uuimzq2]x dx. It contains csc[sup:1uuimzq2]2[/sup:1uuimzq2]x, which is d/dx of (-cot x). At the same time, it can be expressed as (csc x)(cot x)(csc x) dx which contains (csc x)(cot x) which is d/dx for (-csc x). If I use the first possibility for d/dx, the integral evaluates to -(1/2)cot[sup:1uuimzq2]2[/sup:1uuimzq2]x+c. If I use the second possibility for d/dx, it evaluates to -(1/2)csc[sup:1uuimzq2]2[/sup:1uuimzq2]x+c. But cot x is not equal to csc x, so which is the right one to go with?

 

[Deleted member 4993]

I'm trying to find the indefinite integral of cot[sup:3a5v9lmx]3[/sup:3a5v9lmx]x dx. I make the substitution of cot[sup:3a5v9lmx]2[/sup:3a5v9lmx]x = (csc[sup:3a5v9lmx]2[/sup:3a5v9lmx]x -1) and after multiplying it out I end up with 2 integrals. I have a question about one of them, the indefinite integral of cot x csc[sup:3a5v9lmx]2[/sup:3a5v9lmx]x dx. It contains csc[sup:3a5v9lmx]2[/sup:3a5v9lmx]x, which is d/dx of (-cot x). At the same time, it can be expressed as (csc x)(cot x)(csc x) dx which contains (csc x)(cot x) which is d/dx for (-csc x). If I use the first possibility for d/dx, the integral evaluates to -(1/2)cot[sup:3a5v9lmx]2[/sup:3a5v9lmx]x + c[sub:3a5v9lmx]1[/sub:3a5v9lmx]. If I use the second possibility for d/dx, it evaluates to -(1/2)csc[sup:3a5v9lmx]2[/sup:3a5v9lmx]x + c[sub:3a5v9lmx]2[/sub:3a5v9lmx] . But cot x is not equal to csc x, so which is the right one to go with?

[-(1/2)cot[sup:3a5v9lmx]2[/sup:3a5v9lmx]x] - [-(1/2)csc[sup:3a5v9lmx]2[/sup:3a5v9lmx]x]

= [-(1/2)cot[sup:3a5v9lmx]2[/sup:3a5v9lmx]x] + (1/2)[cot[sup:3a5v9lmx]2[/sup:3a5v9lmx]x +1] = 1/2 .............. a constant - which makes for different integartion constant

 

[galactus]

The thing is, \(\displaystyle \frac{-cot^{2}(x)}{2}\) and \(\displaystyle \frac{-csc^{2}(x)}{2}\) only differ by 1/2.

That is taken up in the constant of integration.

i.e.

\(\displaystyle \frac{-csc^{2}(x)}{2}-\left(\frac{-cot^{2}(x)}{2}\right)=\frac{-csc^{2}(x)}{2}+\frac{cot^{2}(x)}{2}\)

\(\displaystyle \frac{-csc^{2}(x)}{2}+\frac{(csc^{2}(x)-1)}{2}\)

\(\displaystyle \frac{-csc^{2}(x)}{2}+\frac{csc^{2}(x)}{2}-\frac{1}{2}=-\frac{1}{2}\)

I even ran this through my calculator for a check, and when I integrated \(\displaystyle cot^{3}(x)\) it returned the cot solution.

When I integrated \(\displaystyle cot(x)csc^{2}(x)\) portion, I got the csc(x) solution.

 

[tor116]

Thanks for the reply. Just so I'm sure I understand, do you mean that either answer is correct, it's just that the integration constants are different? If so, what if this becomes a definite integral? Once the integration constants subtract out wouldn't I still get different solutions depending on whether I had cot[sup:16qlzaxf]2[/sup:16qlzaxf]x or csc[sup:16qlzaxf]2[/sup:16qlzaxf]x?

Thanks again for the reply!

 

[Deleted member 4993]

Thanks for the reply. Just so I'm sure I understand, do you mean that either answer is correct, it's just that the integration constants are different? If so, what if this becomes a definite integral? Once the integration constants subtract out wouldn't I still get different solutions depending on whether I had cot[sup:3sa33nai]2[/sup:3sa33nai]x or csc[sup:3sa33nai]2[/sup:3sa33nai]x?

Thanks again for the reply!

No - you would get the same answer. Why don't you try one for yourself and be convinced.

 

[tor116]

You're right. Thanks!