quick integration questions (1 Viewer)

[FD3S-R]

Find :

integral of 1/ (5+4cos2x) dx

Find :

integral of xcos(x^2) dx


anyhelp appreciated thanks!!

 

[alphatango]

First one -- I can give you an answer (from Mathematica, which I've differentiated and checked to be correct): (-1/3)*Arctan(3*Cot(x)) -- as to how to get to the answer, I can't remember...it's been a while since I did all my 4U techniques.

Second one -- simple substitution, try substituting u=x^2 if you can't see it immediately.

 

[FinalFantasy]

Find :

I=integral of xcos(x^2) dx

let u=x²
du=2x dx
I=1\2 int. cos u du=1\2 sin u+c=1\2 sin x²+c

 

[FinalFantasy]

Find :

I=integral of 1/ (5+4cos2x) dx

let u=2x
du=2dx
I=1\2 int. du\(5+4cos u)
let t=tan (u\2)
du=2dt\(1+t²)
therefore I=1\2int. 2dt\(5+5t²+4-4t²)
=int. dt\(t²+9)
=1\3 tan^-1 t\3+c
=1\3 tan^-1 tan (u\6)+c
=1\3 tan^-1 tan (x\3)+c

 

[FD3S-R]

awesome thanks alpha tango and final fantasy

i just got stuck in the 2nd one cos i didnt know how to express cos2x in terms of t

is it safe to assume to always use substitution to create what ever is next to the x to just cosu, sinu, tanu etc

the first question i was just blind so simple

anyways thanks

 

[FinalFantasy]

awesome thanks alpha tango and final fantasy

i just got stuck in the 2nd one cos i didnt know how to express cos2x in terms of t

is it safe to assume to always use substitution to create what ever is next to the x to just cosu, sinu, tanu etc

the first question i was just blind so simple

anyways thanks

"is it safe to assume to always use substitution to create what ever is next to the x to just cosu, sinu, tanu etc"
i dunno.. i juz looked at the integral and saw dat way, and did it.
i didn't "assume" to do any substitutions heh

 

[alphatango]

It's a very common trick, yes. But it's not always safe to assume that it's what is needed for the question. I don't think there are more than a few pathological cases where it actually makes the integral more difficult, but there are certainly cases where it won't help you see anything better.

 

[underthebridge]

hey I haven't done integration using t formula, but for the second one, could you use the doubl angle result of cos2x, and end up with an inverse tan once integrated?

 

[ngai]

is it safe to assume to always use substitution to create what ever is next to the x to just cosu, sinu, tanu etc

no
actually, stuff like xcosx usually means integration by parts
this time, ur meant to see that the x^2 in cos(x^2) is basically unbreakable, so substituting u=x^2 would be a good idea

 

[ngai]

i just got stuck in the 2nd one cos i didnt know how to express cos2x in terms of t

hey I haven't done integration using t formula, but for the second one, could you use the doubl angle result of cos2x, and end up with an inverse tan once integrated?

why u ppl keep mentioning cos2x?
isn't it cos(x^2)?
and none of the cos2x formulae give cos(x^2)...they give cos^2 (x)

 

[KFunk]

Find :

integral of 1/ (5+4cos2x) dx

I think they're maybe talking 'bout this one. Using the double angle formula there could make things rather interesting...

 

[underthebridge]

yeh thats the one i was talkin about, and with xcos(x^2) i'd use u=x^2, so u end up with integrating 1/2 cos u du, off the top of my head, could be wrong