I need to find the integral from 0 to 1 of the function below. I have tried using integration by parts and then another integration by parts with a substitution, but I am unable to come up with the correct answer.
x[(1-x)^(1/7)] dx
Sorry I cannot make it easier to read.
In an integration by parts, you have a product of two functions, one of which is going to get integrated and the other one differentiated (in the hope that this will simplify the original integral). In general, a function becomes simpler when differentiated, but more complicated when integrated. So in an integration by parts, it's convenient if the function that gets differentiated is one that becomes very much simpler when differentiated. Typical such functions are and . If you see one of these in an integral, then it's worth seeing whether integration by parts will pay off. Otherwise, substitution is likely to be a more useful technique.
Substitution is often unpopular with students, because it's not an automatic process: you have to think about what substitution to make, and that's a skill that only comes with practice. But it's worth persevering with it, because it is definitely the most effective technique for the majority of integrals.