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Integration Under the Integral Sign

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Integration under the integral sign is the use of the identity

int_a^bdxint_(alpha_0)^alphaf(x,alpha)dalpha=int_(alpha_0)^alphadalphaint_a^bf(x,alpha)dx
(1)

to compute an integral. For example, consider

int_0^1x^alphadx=1/(alpha+1)
(2)

for alpha>-1. Multiplying by dalpha and integrating between a and b gives

int_a^bdalphaint_0^1x^alphadx=int_a^b(dalpha)/(alpha+1)
(3)
=ln|(b+1)/(a+1)|.
(4)

But the left-hand side is equal to

int_0^1dxint_a^bx^alphadalpha=int_0^1(x^b-x^a)/(lnx)dx,
(5)

so it follows that

int_0^1(x^b-x^a)/(lnx)dx=ln((b+1)/(a+1))
(6)

(Woods 1926, pp. 145-146).

See also

Integral, Integral Sign, Integration, Leibniz Integral Rule

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References

Woods, F. S. "Integration Under the Integral Sign." §61 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 145-146, 1926.

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Integration Under the Integral Sign

Cite this as:

Weisstein, Eric W. "Integration Under the Integral Sign." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IntegrationUndertheIntegralSign.html

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