We see with the aid of an integration by parts that ......
Theorem 2 makes it legitimate to apply integration by parts.
By $k$-fold integration by parts, ......
This bound is integrable and on integration one again obtains the same form of ......
Functions which are equal almost everywhere are indistinguishable as far as integration is concerned.
Without losing any generality, we could have restricted our definition of integration to integrals over all of $X$. [Not: “Without loosing”]
Recently proofs have been constructed which make no appeal to integration.
To justify the interchange of summation and integration, consider ......
The interchange in the order of integration was legitimate, since ......
One is tempted to reverse the order of integrations but that is illegitimate here.
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