Example:

It seems the following substitution would help:

We do not have the proper expression for dy in the integral, so we rearrange the differential transformation formula to fit better.

The integral is valid on intervals where the denominator is not zero.

We could have also adjusted the integral algebraically: Multiply the numerator by three so that it coincides with the expression for dy, then divide the integral by three to keep the balance. We will show this approach in the next solution.

Example: Look at the following integral. Here the composed function clearly points to the substitution y = 1 − x2. In order to succeed we need to have −2xdx in the integral, so we will create it there even before we start on the substitution. Then we will have to deduce from the basic substitution equality some formula that would allow us to transform the numerator.

Example: Consider the integral

Here the square root seems to bother us. We will try to get rid of it by substitution, but its success is not immediately apparent, because it obviously does not fit well.

We have to try to create a formula for dx, but we have to do it in sucha way that there are only y on the other side, since no x are allowed in the new integral. We will also need a formula for x, but that is simple.

Since we managed to express all parts of the original integral in the language of y, the substitution succeeds:

We obtained an integral of a rational function, which can be evaluated using the partial fractions approach, see this problem in Solved Problems - Integration.

Remark: Sometimes one can simplify calculations quite a bit by starting with algebraic changes right away. In the above example we first deduced a transformation formula for differentials and then pulled dx out of it. It is much easier if we reverse the order: First solve the basic substitution equality for x (which in fact makes this into an indirect substitution) and then obtain the necessary formula for dx by differentiation.

Example: Consider the integral

Here the expression would get markedly better if there was just a letter instead of those exponentials. This suggests the substitution y = ex. I would be really nice if we had exdx in the integral, but it is not there, so we need to deduce a formula for it. We try it both ways, first throught the formula for differentials, then throught an indirect substitution.

It worked, so

This is an integral that can be easily evaluated using partial fractions. It is useful to remember that with an exponential substitution we never have problem with dx.

Example: Consider the following problem.

Here we would like to get rid of trig functions, but first we have to decide whether we should substitute for sine or for cosine. We know that if we use y = sin(x), then we would need cos(x)dx, whereas the substitution y = cos(x) requires -sin(x)dx. Is theree a chance to create something like this? Obviously the only source is the numerator, from there it would be really hard to extract sine, but there is a way to get cosine out of it. This points to the sine substitution, but then we will need also a formula for the cosine in the denominator.

The substitution went through and the new integral is easy to handle using partial fractions decomposition.