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# Basic standard integral

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1. Basically we have been given a sheet with all the standard integrals we should know.

I don't however understand this one:

Integral f'(g(x))g'(x)dx = f(g(x)) + c

Would someone using latex just give an example using a simple equation as then I should see it much clearer!

Thanks
2. (Original post by TimothyTankT)
Basically we have been given a sheet with all the standard integrals we should know.

I don't however understand this one:

Integral f'(g(x))g'(x)dx = f(g(x)) + c

Would someone using latex just give an example using a simple equation as then I should see it much clearer!

Thanks
Use a substitution of and it should drop out. That integrand is in the form of the derivative of a composite function.
3. (Original post by TimothyTankT)
Basically we have been given a sheet with all the standard integrals we should know.

I don't however understand this one:

Integral f'(g(x))g'(x)dx = f(g(x)) + c

Would someone using latex just give an example using a simple equation as then I should see it much clearer!

Thanks
It is the rule for the integration of compodite functions and the base of this notation
is the definition of undetermined integral namely
so that

from this

If function f in composite, for example f(g(x)), then from the chain rule
where f' is derivative of outer function
substituting g(x) in x multiplied by the derivative of inner function, g'(x).
So integrating this

Generally
where H is the primitive function of h.
For example:

because the primitive function of 1/x is ln|x| and the derivative of cosx is -sinx.
F.e 2.

4. Try differentiating f(g(x)). Use a substitution u = g(x).

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Updated: October 28, 2010
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