Why is there no general rule for integrating the product of two functions?

[O.J.]

is there a general rule to integrate th eproduct of two functions? i aksed my prof and he said no gen rule exists. and i really wonder why... how come differentiation has a general formula for a derivative of a product of 2 functions while integrations doesnt

 

[Galileo]

Since differentiation and integration are in a way inverse processes, every rule for differentiation has corresponding rule for integration.
Just write down the product rule for differentiation and integrate both side of the equation. There's your very useful rule for integrating a product. It's called integration by parts.

 

[O.J.]

lets take one of the parts of the equation. d(uv)/dx=u(dv/dx)+v(du/dx)
now, let's integrate: left part is easily uv, right part: how will u know how to integrate u(dv/dx).

another question: what I am asking about is how to INTEGRAE THE PRODUCT OF TWO FUNCTIONS WITH RESPECT TO X, not how to integrate the derivative of the product of 2 functions...

 

[HallsofIvy]

Yes, we understood that. Galileo gave you the answer: integration by parts! Go back and read what he said again.

 

[Tomsk]

Integration by parts works when you want to integrate something of the form u*dv/dx, where u is easily differentiated, and dv/dx and v*du/dx are easily integrated. So it's not a completely general rule, but it is very useful. You can repeat the process if v*du/dx isn't easily integrated as well.

 

[Tomsk]

That's probably really unclear, an example should help.

$$\int{u \frac{dv}{dx} dx} = u v - \int{v \frac{du}{dx} dx}$$

I think you'd got there. Now try for example u=x, dv/dx = e^x:

$$\int{x e^{x} dx} = x e^{x} - \int{e^{x} 1 dx}$$
$$\int{x e^{x} dx} = x e^{x} - e^{x} + C= e^{x}(x-1) + C$$

And there's your answer. You can stick limits in too, they don't change. You have to evaluate uv at those limits too.