Integration Questions (2 Viewers)

[KFunk]

for 48.

is the next line -(n-1) int. (sinxcos^(n-2)x) dx??

I think it should be (positive) (n-1) ∫ cos^n-2xsin²xdx

 

[KFunk]

46. use integration by parts:

∫ u'v = uv - ∫ uv' where u' = sinxcos²x and v=sin^n-1x

When integrating sinxcos²x use the substitution u= cosx

See if you can approach the form I_n = (n-1)/3.I_n-2 - (n-1)/3.I_n

 

[Slidey]

LOL. Yeah I thought there was an x there.

integral (square root of (16 + x^2)

t=x+sqrt(x^2+16)
(t-x)^2=x^2+16
t^2-2xt+x^2=x^2+16
t^2-16=2xt
x=(t^2-16)/2t
So sqrt(x^2+16)=t-x=(t^2+16)/2t
dx=(t^2+16)/2t^2 dt

∫ sqrt(x^2+16) dx=1/4 ∫ (t^2+16)^2/t^3 dt
1/4 ∫ (t^4+32t^2+256)/t^3 dt
1/4 ∫ t+32/t+256t^(-3) dt
1/4(t^2/2+32ln(t)-256/2t^2) + C
Now,
t=x+sqrt(x^2+16)
t^2=x^2+2sqrt(x^2+16)+x^2+16
t^2=2x^2+16+2sqrt(x^2+16)
256/t^2=2x^2+16-2xsqrt(x^2+16) (rationalising)
So,
∫ sqrt(x^2+16)dx = 1/4(t^2/2+32ln(t)-256/2t^2) =
1/8(2x^2+16-2xsqrt(x^2+16)+2x^2+16+2xsqrt(x^2+16)+8ln(x+sqrt(x^2+16) + C =
(xsqrt(x^2+16)+16ln(x+sqrt(x^2+16))/2 + C

 

[richz]

for 48 is is sin^2 x because u have to derive cosx too??

 

[Slidey]

yeah sliderule theres no x
substitute x = 4tanp

Or that. Why do I always take the long way.

 

[richz]

slide, now that sounds better.

 

[richz]

lol , i knew u had to let u= sinx or cos x but i didnt know it was u=4tanx

 

[Slidey]

Oh, so the algebra bash produced the same answer as the trig attack? That doesn't usually happen. Cool.

 

[KFunk]

for 48 is is sin^2 x because u have to derive cosx too??

yeah man, since you're using int. by parts:
∫ u'v = uv - ∫ uv' where u' = cosx and v=cos^n-1x

you have to integrate cosx which = sinx
and differentiate cos^n-1x which = (n-1)cos^n-2x(-sinx)

combine these and you get [- ∫ uv'] = (n-1) ∫ cos^n-2xsin²xdx

 

[richz]

thnx kfunk for 48

 

[richz]

lol, slide i didnt know how u did it i would just use shaf's one

 

[KFunk]

no problem, I think the gist of 46 is back there somewhere as well. This thread has become quite chaotic.

 

[Slidey]

lol, slide i didnt know how u did it i would just use shaf's one

Yeah, that's fine. That's a far more elegant way.

 

[richz]

ya thnx anyway for ur help