Find the following integrals:
cosec^2x / (2 + cotx)^3 . dx
the answer starts with
let y = (2 + cotx)^ -2
my question is how did they know that y should be equal to (2 + cotx )^ -2
thankyou.
cosec^2x / (2 + cotx)^3 . dx
the answer starts with
let y = (2 + cotx)^ -2
my question is how did they know that y should be equal to (2 + cotx )^ -2
thankyou.
-
I will answer your question first
They know that the substitution must be (2+cotx)^(-2) because when this is differentiated it will generate a power of -3. Since you see a power of +3 in the integral it means that there is probably some canceling out as we will see!
y=(2+cotx)^(-2) dy=(-2)(2+cotx)^(-3)(-1/sin^2x)dx (the term -1/sin^2x comes from differentiating
cotx)
Substitute these in the original integral bearing in mind that since y=(2+cotx)^(-2) you sqyare both sides to get :
to get y^2=2+cotx
Integral((( sin^2x)sin^2x)/(y^6)*2*(2+cotx)^(-3))dy=…
and going back to the x 1/2(2+cotx)^(-2)+C
They know that the substitution must be (2+cotx)^(-2) because when this is differentiated it will generate a power of -3. Since you see a power of +3 in the integral it means that there is probably some canceling out as we will see!
y=(2+cotx)^(-2) dy=(-2)(2+cotx)^(-3)(-1/sin^2x)dx (the term -1/sin^2x comes from differentiating
cotx)
Substitute these in the original integral bearing in mind that since y=(2+cotx)^(-2) you sqyare both sides to get :
to get y^2=2+cotx
Integral((( sin^2x)sin^2x)/(y^6)*2*(2+cotx)^(-3))dy=…
and going back to the x 1/2(2+cotx)^(-2)+C