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Diptanuj Sarkar

1 month before

Euler Substitutions

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I came across the Euler Substitutions when I was reading the theory for Indefinite Integration. But I cannot understand the motivation behind these substitutions, i. e. how we arrive at these substitutions, and why the cases. Can someone please explain them to me, or point me to some resource that does?


Amrita Roy

1 month before

Hi Diptanuj, In general, we have root ax^2+bx+c accompanied by function of the form px+q..In those cases the general idea is to express px+q as a combination of the derivative of the quadratic and a constant. This enables you to break the given integral in two standard parts. This is the general idea behind this manipulation.. However when you have f(x, root ax^2+bx+c) as shown in the example, it is difficult to replace the x in terms of t if you have some general substitution like ax^2+bx+c=t etc. I am attaching an image to show you how this first substitution helps you to reduce it down to some general form. Let me know if you have any problems in those steps. But if you understand them, try the other two substitutions to understand the essence of this theory.
Amrita Roy

1 month before

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However, as mentioned in the Vedas, this can lead to some critical calculations.So, use this only when you don't see any other method working. In examination, sometimes it is mentioned that you have to use this substitution. You can see this kind of application in passage based questions.