Integration using parametric derivatives - Alchetron, the free social encyclopedia

In mathematics, integration by parametric derivatives is a method of integrating certain functions.

For example, suppose we want to find the integral

∫ 0 ∞ x 2 e − 3 x d x .

Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it. However, we may also evaluate this by starting with a simpler integral and an added parameter, which in this case is t = 3:

∫ 0 ∞ e − t x d x = [ e − t x − t ] 0 ∞ = ( lim x → ∞ e − t x − t ) − ( e − t 0 − t ) = 0 − ( 1 − t ) = 1 t .

This converges only for t > 0, which is true of the desired integral. Now that we know

∫ 0 ∞ e − t x d x = 1 t ,

we can differentiate both sides twice with respect to t (not x) in order to add the factor of x² in the original integral.

d 2 d t 2 ∫ 0 ∞ e − t x d x = d 2 d t 2 1 t ∫ 0 ∞ d 2 d t 2 e − t x d x = d 2 d t 2 1 t ∫ 0 ∞ d d t ( − x e − t x ) d x = d d t ( − 1 t 2 ) ∫ 0 ∞ x 2 e − t x d x = 2 t 3 .

This is the same form as the desired integral, where t = 3. Substituting that into the above equation gives the value:

∫ 0 ∞ x 2 e − 3 x d x = 2 3 3 = 2 27 .

References

Integration using parametric derivatives Wikipedia

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