Assumptions (2) 

For certain "ParameterTreatment" settings, ParameterizedFunctionIntegrate will return conditions on parameters that ensure convergence:

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With Assumptions, a result valid under the given assumptions is returned:

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It is generally more appropriate and effective to use Assuming[assump,ParameterizedFunctionIntegrate[…]] rather than ParameterizedFunctionIntegrate[…,Assumptions→assump], as the latter only applies assumptions when computing a generic antiderivative with Integrate, while the former applies assumptions at all stages of the computation.

GenerateConditions (2) 

For certain "ParameterTreatment" settings, ParameterizedFunctionIntegrate will return conditions on parameters that ensure convergence:

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Use GerenateConditions→False to suppress conditions in these cases:

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The GenerateConditions option is only effectual when the "ParameterTreatment" option is set to "Generic" or "Conditional".

GenerateConditions→False can speed up integration:

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MostGeneralFirstQ (2) 

By default, the first clause in Piecewise results returned by ParameterizedFunctionIntegrate is the one that applies most generically with respect to specializaiton of parameter values:

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The opposite ordering, with the most generic clause given last, can be obtained using MostGeneralFirstQ→False. This gives predicates that may be easier to understand at a glance:

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ParameterTreatment (5) 

Using "ParameterTreatment" → "Generic" gives the most generic result, equivalent to the result of Integrate:

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However, this generic antiderivative quietly returns an incorrect antiderivative when later specialized to α → -1:

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For generic antiderivatives, substitution does not necessarily commute with integration:

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ParameterTreatment→"Conditional" attempts to indicate what specializations of a generic antiderivative are valid with ConditionalExpression:

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The result of an invalid substitution is now more appropriately given as Undefined rather than ComplexInfinity:

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By default, the built-in Wolfram Language Integrate already returns ConditionalExpressions when appropriate for parametric definite integrals:

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But not in the coresponding indefinite case:

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For antiderivatives, ParameterizedFunctionIntegrate[…,"ParameterTreatment"→"Comprehensive"] returns a Piecewise function that is valid for all combinations of parameter values. For parametric integrands, the result is often a Piecewise result wherein one of the consequents is a generic antiderivative :

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Specialization gives a correct result for all α:

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The following plot illustrates the discontinuity of this Piecewise result at the parameter value α=-1:

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At the expense of more computing time, "ParameterTreatment" → "Continuous" attempts to return a Piecewise result that is maximally continuous:

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The generic first consequent is different from the corresponding in the generic consequent of the merely "Comprehensive" result. To verify that consequent is valid:

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The different consequent makes the overall antiderivative continuous with respect to parameter α, as suggested by the following plot:

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The parametric continuity was accomplished by introducing a particular integration constant that is a function of at most the parameter alone, making its partial derivative with respect to the integration variable z vanish. Although parametric continuity is not necessary for some applications, including definite integration, see "Applications" for a differential equation example where parametric continuity is necessary.

At the expense of more time, and perhaps also more catastrophic cancellation when substituting floating-point numbers near a special case, setting "ParameterTreatment"→"ApartContinuous" will attempt to make the presence of intentionally-introduced constants more obvious in a separate term:

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Contrast with the merely "Continuous" result:

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PrincipalValue (2) 

Certain ordinary Riemann definite integrals are divergent:

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By interpreting instead as a Cauchy principal value integral, we can give meaning to such cases:

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TrueInPiecewiseMeans (3) 

Use "TrueInPiecewiseMeans"→ All to Print a user assistance message which expands the True predicate in Piecewise results into a condition on parameters:

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By default, "TrueInPiecewiseMeans"→False and no messages are printed:

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When "TrueInPiecewiseMeans"→ Automatic, a message is printed when the number of Piecewise clauses is greater than two:

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But not otherwise:

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When can continuity matter? (4) 

DSolve returns generic general solutions to ordinary differential equations without boundary conditions:

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But the specialization of the particular portion of the above generic solution for the resonance as ω → κ is incorrect:

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We can instead compute the particular portion of the generic general solution with parametrically-continuous comprehensive antiderivatives, and a result valid at resonance is obtained. The first clause of this Piecewise result is the generic particular solution. The second clause is its specialization to the particular solution for ω = κ:

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Using this method with the built-in Integrate misses the discontinuity at ω = κ:

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