Integrals of the form dx are calculated with the aid of the three Euler substitutions.
Ex.1 Evaluate I =
Sol. In this case a < 0 and c < 0 therefore neither the first, nor the second, Euler substitution is applicable.
But the quadratic trinomial 7x – 10 – x^{2} has real roots α = 2, β = 5, therefore we use the third Euler substituion :
Ex.2 Evaluate
Sol.
...(1)
...(2)
We know
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1. What are Euler's substitutions? 
2. How do Euler's substitutions work? 
3. What are some common examples of Euler's substitutions? 
4. Can Euler's substitutions be used for all types of differential equations? 
5. Are there any limitations or drawbacks to using Euler's substitutions? 

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