Example: Consider the integral

We try the substitution y = x² + 1 that seems to be offering itself. There is x missing in the integral, but it won't be very nice in this case.

Thus we get

which does not look any better than the original problem, perhaps even worse. So this particular substitution did not help and other substitutions are not apparent.

Example: Consider the integral

We try this substitution:

We get

and the substitution did not help, creating x turned out to be too complicated. No method covered in Math Tutor would help with this new integral.

Example: Consider the integral

Here we have x in the denominator, which just begs for the substitution y = ln(x):

Although this integral looks somewhat better than the original one, we still do not know how to evaluate it - the sine function still spoils it and there is no way to get rid of it via substitution.

Remark: The difference between success and failure can be very small. On the page with examples of successful substitutions we had the integral

which we easily solved by denoting the whole denominator by y. Now we make a little change:

The expression next to dx differs by one from the numerator and this missing one can not be fixed. The only hope would be to write

and then substitute for x by solving the basic substitution equality; however, this we cannot do (we do not know how to solve polynomials of the third degree using a nice formula). Thus for this integral substitution does not help, just because a one instead of a two at the wrong place.