How to integrate v^(3v^2 - 1) dv ?
Thanks.
Originally posted by Snoopyies:How to integrate v^(3v^2 - 1) dv ?
Thanks.
Cannot integrate analytically. Use numerical methods such as Simpson's rule.
I go google for Integration Simpson's rule and the rule says that it only applies to find the definite integration ie with the limits, the rule cannot be applied to find indefinite integration.
By the way, how to tell whether an integration question can be integrated or cannot be integrated ? Are there any rules or guidelines on it ?
The substition method in integration is used to reverse the differentiation chain's rule and the integration by parts in integration is used to reverse the differentiation product rule.
Is there a method in integration that can be used to reverse the differentiation quotient rule ?
Is there also a method in integration that can be used to reverse the implicit differentation ?
No hard and fast rule. Basically consider the following methods.
1. Substitution
2. Parts (This is the reverse of the product rule)
3. Trigonometric Substitution
4. Partial Fractions
If all the above fails then unlikely you can integrate analytically. Again you can try chim methods like expanding the function into a Taylor Series (but you must check for convergence) and integrate term by term.
No. I am not aware of any method that reverses the quotient rule or implicit differentiation.
Do you want to integrate v^(3v^2-1) or v*(3v^2-1). The latter case is easy but I still can't think of a way to integrate the former case.
Mathematica & Maple turned up blank. I don't think this is directly integrate-able.
Originally posted by Snoopyies:I go google for Integration Simpson's rule and the rule says that it only applies to find the definite integration ie with the limits, the rule cannot be applied to find indefinite integration.
By the way, how to tell whether an integration question can be integrated or cannot be integrated ? Are there any rules or guidelines on it ?
The substition method in integration is used to reverse the differentiation chain's rule and the integration by parts in integration is used to reverse the differentiation product rule.
Is there a method in integration that can be used to reverse the differentiation quotient rule ?
Is there also a method in integration that can be used to reverse the implicit differentation ?
To TS,
Case 1
Using Substitution Method in integration to reverse the differentiation chain's rule
Differentiation by chain's rule
Let y = (3x^2 - 1)^3
dy/dx = 3(3x^2 - 1)^2 . (6x)
= 18x(3x^2 - 1)^2
Integration by Substitution Method
Integrate 18x(3x^2 - 1)^2 dx
Let u = 3x^2 - 1
du/dx = 6x
dx = du/6x
Substitute u = 3x^2 -1 and dx = du/6x into
Integrate 18x(3x^2 - 1)^2 dx
= Integrate 18x u^2 . du/6x
= Integrate 3u^2 du
= 3(u^3)/3 + c
= u^3 + c
= (3x^2 - 1)^3 + c since u = 3x^2 - 1
PS :
Case 1
Differentiation by chain's rule
Let y = g(u) where u = f(x)
dy/du = g'(u) du/dx = f '(x)
dy/dx = g'(u). f '(x)
Integration by substitution method
Integrate (dy/dx) dx = Integrate g'(u).f '(x) dx
= Integrate g'(u) (du/dx) dx
= Integrate g'(u) du
y = g(u)
Case 2
Using Integration by parts to reverse the differentiation product rule
Differentiation by Product Rule
y = uv
dy/dx = u (dv/dx) + v (du/dx)
Integration by parts (for product rule)
Integrate (u) (dv/dx) dx = uv - Integrate v (du/dx) dx
Integrate (u) (dv/dx) dx + Integrate v (du/dx) dx = uv
Integrate [ (u)(dv/dx) + v (du/dx) ] dx = uv
Case 3
Integration by parts to reverse differentiation by quotient rule
Differentiation by Quotient Rule
Let y = u/v where u = f(x) v = g(x)
du/dx = f '(x) dv/dx = g'(x)
dy/dx = [ v.du - u.dv ] / v^2
Integration by Parts (for quotient rule)
Integrate (du/v) dx = u/v + Integrate (u/v^2.dv) dx
Integrate (du/v) dx - Integrate (u/v^2.dv) dx = u/v
Integrate (v.du) / v^2 dx - Integrate (u.dv)/v^2 dx = u/v
Integrate [ v.du - u.dv ] / v^2 dx = u/v
Relationship between Integration by parts (for product rule) and integration by parts (for quotient rule)
y = u/v can be easily re-expressed as y = uv^-1
and so the integration by parts (for product rule) can be used instead of the integration by parts (for quotient rule)
umm... so what is the answer? or are you just showing the different methods of integration?
Originally posted by ThunderFbolt:umm... so what is the answer? or are you just showing the different methods of integration?
Snoopyies's question
Integrate v^(3v^2 -1) dx
is unlikely to have a unique function as the integration answer.
Hence, when you entered the function into Mathematica and maple or wolfram, you do not get an answer.
When we cannot get a unique function in the integration answer, we will need to use definite integration ie to use limits and then we can use the numerical approximate methods of integration eg Simpson's rule to find it.
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I am answering to the other questions raised by Snoopyies.
"The substitution method in integration is used to reverse the differentiation chain's rule and the integration by parts in integration is used to reverse the differentiation product rule.
Is there a method in integration that can be used to reverse the differentiation quotient rule ?
Is there also a method in integration that can be used to reverse the implicit differentation ?"
The last two questions are interesting as we do not see any integration method to reverse differentiation quotient rule and implicit differentiation in maths textbooks.
I am still thinking whether there is a method in integration that can be used to reverse the implicit differentiation.