What method to calculate intergration?

[HeilPhysicsPhysics]

For example
Intergrate (uv dx)=?
Intergrate (u dx/v)=?
Intergrate (u^v dx)=?
Intergrate (log_u v)=?
Where u and v are the function of x.
Is there any method in intergration just like dy/dx=(dy/du)(du/dx) in diffrentiation?

 

[d_leet]

Intergrate (uv dx)=?

Integration by parts is the general method for this.

Intergrate (u dx/v)=?

This can be seen as a special case of the first.

Intergrate (u^v dx)=?

You'll probably run into problems with this one most often where there won't be an integral in terms of elementary functions. There is no general method, and I don't think I've really seen too many integrals like this.

Intergrate (log_u v)=?
?

This is again a special case of the first.

Is there any method in intergration just like dy/dx=(dy/du)(du/dx) in diffrentiation?

What is essentially this property in reverse is often used to solve integrals and it is usually called integration by substitution.

 

[Data]

All the rules you've learned for differentiation have counterparts for integration. But integration techniques are often more difficult to use. This is because not all elementary functions have antiderivatives that are expressible in terms of elementary functions, and finding out which ones do isn't always easy. In contrast, all the elementary functions have derivatives which are also expressible in terms of elementary functions. In some way it also is usually more difficult to see what operations will allow you to find antiderivatives, even when they do exist in terms of elementary functions, than it is for derivatives, at least when you are just starting to do them (for example, when you want to integrate a product of functions, you can try to use integration by parts, but sometimes trying this will result in another integral that seems harder - and so you have to go back and try something else).

(when I talk about elementary functions, I mean things like polynomials, trig [and inverse trig] functions, exponentials, logarithms, and quotients, products, sums, differences, roots, and compositions of these)

 

[jbusc]

doing a change of variables is about the most useful integration technique, I've found (though depending on your application, integration by parts might be more necessary)

You don't even need to know a lot of complex subsitutions. Polar, cylindrical, spherical coordinates all are incredibly useful.

 

[eljose]

The best formula ever for integration (Bernoulli)

$$\int dxf(x)=C+\sum_{n=0}^{\infty}(-1)^{n} x^{n+1}\frac{1}{n!} \frac{d^{n} f}{dx^{n}}$$ where C is a constant.