Choosing the Right u to Integration by Substitution

[sporus]

i don;t have a specific homework question. i have a sort of conceptual question instead

when integrating by substitution, how do i know what to choose as u?

for example


integral of z^2 / (1 + z^3)^(1/3) dz

i am suppposed to choose u as 1+z^3. any other value for u won't give me the right answer. so my question is how do i know what to choose as u? i could have chosen something else but how do i choose the right one?

 

[fzero]

any other value for u won't give me the right answer.

It's not that other choices won't give the right answer, it's that other choices might not lead to integrals that you can do. The trick to identifying substitutions is having a firm grasp of what integrals can be solved in terms of elementary functions. You then look for substitutions that put the integral in question into one of those forms. There aren't really a set of rules to follow, it's something that comes about from experience.

However, your example is of the form

\int f(z) g(z) dz,

where g(z) can be observed to be proportional to f'(z). This type of factorization is one of the first things you look for.

 

[bopll]

right, this is definitely something that comes with experience. You'll notice patterns that will help you choose u. For this example, since you know you will get (something)x^2 when you take the derivative of (something)x^3, you know that the term with the x^3 will want to include u. You include the 1+ because it turns a binomial into one term, which is much easier to integrate.

Most Calc 1 problems use simple patterns like these... Generally when you have any polynomial in parentheses raised to some power, you will want to see if you can set what's inside the parentheses as u first.

 

[sporus]

ok, thanks