integration 2

[gutswarrior]

(csc^4) x / (cot^2) x

 

My answer is 2cot2x + c and the real answer is negative of that.

 

What's the solution? using u substitution

 

Thanks...

[ragnarok1945]

first off, csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x)

 

That means csc(x)/cot(x) = 1/cos(x)

 

Therefore, begin by simplifying the fraction first.

[gutswarrior]

i dont understand... can you apply it to problem

[ragnarok1945]

look, since csc(x)/cot(x) = 1/cos(x), that means csc^2 (x)/cot^2 (x) = 1/cos^2 (x)

 

So now the fraction simplifies down to csc^2 (x)/cos^2 (x)

 

Since csc(x) = 1/sin(x), that means csc^2 (x) = 1/sin^2 (x)

 

That means the fraction can now be turned into 1/[sin^2 (x)*cos^2 (x)]

 

NOW apply sub:

 

Remember, d/dx of sin(x) = cos (x), and d/dx of cos(x) = -sin(x)

[gutswarrior]

how it become cot 2x?

[ragnarok1945]

did you first reduce the fraction like I said to do, before doing the sub?

[gutswarrior]

it become csc^2 sec^2

[ragnarok1945]

yes, but csc^2 = 1/sin^2 and sec^2 = 1/cos^2, so what I said is correct as well.

 

As for the reason for becoming cot(2x), that's a tan/cot property

[gutswarrior]

what should i do now?

[ragnarok1945]

like I said, first off you have to transform the fraction into form I wrote above, then use substitution.

[gutswarrior]

if i use sin x as u there is no cos x above

[ragnarok1945]

yes there is. You said it becomes csc^2(x)*sec^2(x), and that is the same thing as 1/[sin^2(x)*cos^2(x)]

[gutswarrior]

i dont really understand

[ragnarok1945]

well, once that fraction has been transformed into that state, I guess you can separate it into 1/sin^2(x) * 1/cos^2(x). Then you can solve them one part at a time.

 

Oh and as for cot(2x) and all the other properties, trying reading these:

 

http://en.wikipedia.org/wiki/List_of_trigonometric_identities

http://en.wikipedia.org/wiki/Cotangent

[gutswarrior]

this is what i do?

 

(csc^2 x * csc^2 x) / cot^2

(1 + cot^2 x) (csc^2 x) / cot ^2 x

(csc^2 x + cot^2 x*csc^2 x) / cot^2 x

csc^2 x / cot^2 x + (cot^2 x*csc^2) / cot^2 x

- (-csc^2 x) / cot^2 x + csc^2 x

- u^-1 / -1 + (-cot x) + c

1 / cot x - cot x + c

(1- cot^2 x) / cot x * 2/2 + c

2 ((1-cot^2 x) / 2cot x) + c

2cot 2x +c

 

see its positive but the answer is negative...

do i made a mistake?

[ragnarok1945]

- u^-1 / -1 + (-cot x) + c

1 / cot x - cot x + c

 

I think it should be 1/cot(x) + cot(x) + c

 

If I'm wrong then maybe the solution is wrong

[gutswarrior]

that -1 after / is for u, cot x should be negative

integral of csc^2 x is -cot x

 

can you spot mistakes?

[ragnarok1945]

not exactly. You might be doing it right, just the solution is wrong (it happens every now and then)

[gutswarrior]

is my signs for identities are correct?

[ragnarok1945]

I think they are (haven't seen trig identities for a LONG time though), and in any case you should be able to check with the links I gave you

[gutswarrior]

i check my solutions many times but unable to find the errors...

[ragnarok1945]

and I told you there are times where the solution manual is wrong.

[Cyber Altair]

Now that's why I left Maths =\

 

Can't you two continue this over pm? Reduces the spam and chances of off-topic posts and such =P

[ragnarok1945]

Oh I think we're done here, thanks for the advice though

[gutswarrior]

can you explain me back substitute and give example