How would you integrate a function like f(x)=x(1-x)^6?

For a question like this, it would be far too time-consuming to expand the bracket and then multiply through by x. It might then seem that integration by parts is the optimal solution, but this is actually not necessary.

If you make the substitution u=1-x then you have du=-dx. Then the integral of x(1-x)^6 with respect to x is equivalent to the integral of -(1-u)u^6 with respect to u. By multiplying through, we have u^7-u^6 which is not difficult to integrate.

An important thing to remember in this is always to remember to find dx in terms of du - do not assume that dx=du because it rarely does. Finally, make sure you sub x back into your final answer.

If you are given a definite integral where you need a substitution, always remember to change your limits appropriately. There is then no need to sub x into your integral at any point in the working