Is there a general formula for the integration of a composite function?

[Al-Layth]

We have the chain rule in differential calculus. My question is there a corresponding rule in Integral calculus to tell us the general result of the integral:
[math]\int{f(g(x))} dx[/math]in terms of the functions [imath]f[/imath] and [imath]g[/imath]

I tried googling but only found articles and videos on substitution so I came here
thanks

 

[lex]

I'm afraid not.

 

[Dr.Peterson]

We have the chain rule in differential calculus. My question is there a corresponding rule in Integral calculus to tell us the general result of the integral:
[math]\int{f(g(x))} dx[/math]in terms of the functions [imath]f[/imath] and [imath]g[/imath]

I tried googling but only found articles and videos on substitution so I came here
thanks

In fact, substitution is the integration method corresponding to the chain rule! That's probably what you found on your search.

In general, integration, as the inverse of differentiation, requires trial and error, much as division (the inverse of multiplication), and factoring (the inverse of distribution) do. There's a good reason we teach differentiation first.

 

[Al-Layth]

In fact, substitution is the integration method corresponding to the chain rule! That's probably what you found on your search.

In general, integration, as the inverse of differentiation, requires trial and error, much as division (the inverse of multiplication), and factoring (the inverse of distribution) do. There's a good reason we teach differentiation first.

thank you, i did some googling and found that the substitution rule only works in the case of

[math]\int{f(g(x))g'(x)dx}[/math]
but not for:
[math]\int{f(g(x))}[/math]in general?

 

[Al-Layth]

I'm afraid not.

dang

 

[topsquark]

It's (relatively) easy to find the derivative of just about anything. It's usually harder to find an integral.

-Dan