Harder Integration (1 Viewer)

[Dreamerish*~]

I have a few integration questions.

i) ²₀∫ √(4 - x²) dx, x = 2sinθ

ii) ∫ dx/[x(log_ex)²]

Integration kings (and queens), this is your time to shine.

 

[rama_v]

1)

x = 2sin@
dx/d@ = 2Cos@
Int = (from 0 to pi/2) sqrt(4 -4sin²@) . 2Cos@ d@
= 2Cos@.2Cos@.d@
= 4 Int(from 0 to pi/2) Cos²@ d@
= 4 Int (from 0 to pi/2) [(1/2) + (1/2)Cos2@]d@
= 4[(1/2)@ + (1/4)sin2@]^pi/2₀
= 4. pi/4
= pi

 

[acmilan]

i) ∫ √(4 - x²) dx, x = 2sinθ -> dx = 2cosθ dθ

= ∫√(4 - 4sin²θ) 2cosθ dθ
= ∫2√(cos²θ) 2cosθ dθ
= 4∫cos²θ dθ

Just ordinary trig integration from there

ii) ∫ dx/[x(log_ex)²]

Let u = log_ex, du = dx/x

∫ du/u²

iii) ∫ [(log_ex)²]/x dx

Let u = log_ex, du = dx/x

∫u² du

 

[rama_v]

ii)

Let u = ln x
du.dx = 1/x
therefore Int = du/u²
= u^-2du
= u^-1/-1 du
= -1/u + C
= -1/lnx + C

I think thats right but not 100%

 

[Dreamerish*~]

Thanks guys. <3

 

[haboozin]

I have a few integration questions.

i) ²₀∫ √(4 - x²) dx, x = 2sinθ

if the question has "or otherwize" in it.... heres a muc easier way:

this is area of a quarter of a circle radius 2

1/4pi x r^2 = 1/4 pi 4 = pi units²

 

[dawso]

if the question has "or otherwize" in it.... heres a muc easier way:

this is area of a quarter of a circle radius 2

1/4pi x r^2 = 1/4 pi 4 = pi units²

or even if it doesnt hav otherwise, u can draw a parametric circle and solve it in a similar fashion

 

[Sepulchres]

The question specifies the substtution to be made. I wouldnt think you;d get the full marks if you did it "otherwise".

 

[justchillin]

In HSC if there's another way 2 do the quetion... they'll usually cover their bases and say "otherwise"...

 

[Stefano]

So where's the harder integration ?

 

[acmilan]

So where's the harder integration ?

In 3 unit you are rarely (if ever) given trig substitution or integrals in which what you have to substitute isnt given to you.

 

[Stefano]

In 3 unit you are rarely (if ever) given trig substitution or integrals in which what you have to substitute isnt given to you.

Oops, thought these were the Ext.2 forums
You are quite right acmilan.

 

[Dreamerish*~]

Oops, thought these were the Ext.2 forums
You are quite right acmilan.

Haha, Extension 2 is too cool for Dreamerish.