Euler Substitution of the Second Kind
Krystian Karczyński
Founder and General Manager of eTrapez.
Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.
He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.
Euler Substitution of the First Kind (for a>0) – Review
In the previous post:
Euler Substitution of the First Kind
we dealt with integrals of the type:
,
where a>0.
We also solved an example integral that meets this condition, i.e.
But what if
Then the second type of Euler substitution might help us (but it might not…):
Euler Substitution of the Second Kind (for c>0)
Having an integral of the type:
where c>0, we use a substitution of the type:
which we again square on both sides, where this time the terms with
We substitute all this into the integral:
and we end up again with a rational integral, which – I repeat – is generally tedious.
Let’s proceed with an example.
Example
In the quadratic trinomial, the order of terms is slightly changed, but it’s clear that
We substitute:
We square both sides:
The 2 term cancels out (as it should):
And now something that wasn’t in the first kind of substitution, we divide both sides by x:
Next we solve for x:
We have x solved in terms of the variable t. Now we solve for
We only have to solve for
We have thus solved:
, all using the variable
and substitute:
Let’s clean up:
Returning to the substitution:
\frac{1}{\sqrt{2}}\ln \left| u \right|+C=\frac{1}{\sqrt{2}}\ln \left| 1-2\sqrt{2}t \right|+CWe still need to return from t to x. Our Euler substitution was
xt+\sqrt{2}=\sqrt{2+x-{{x}^{2}}}From which
t=\frac{\sqrt{2+x-{{x}^{2}}}-\sqrt{2}}{x}Thus our solution is
\frac{1}{\sqrt{2}}\ln \left| u \right|+C=\frac{1}{\sqrt{2}}\ln \left| 1-2\sqrt{2}t \right|+C=\frac{1}{\sqrt{2}}\ln \left| 1-2\sqrt{2}\frac{\sqrt{2+x-{{x}^{2}}}-\sqrt{2}}{x} \right|+CWhat about other cases?
We know that when in the integral:
- a>0 – we use the first kind of substitution
- c>0 – we use the second kind of substitution
But what if neither
…we’ll choose one of the three kinds of substitution.
Are you looking for college or high school math tutoring? Or maybe you need a course that will prepare you for the final exam?
We are "eTrapez" team. We teach mathematics in a clear, simple and very precise way - we will reach even the most knowledge-resistant students.
We have created video courses translated in an easy, understandable language, which can be downloaded to your computer, tablet or phone. You turn on the video, watch and listen, just like during private lessons. At any time of the day or night.