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 – x2 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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