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The definite integral is called improper if at least one of two conditions is met:

One (or both) of integration limits is equal to or . In this case, the integral is called improper integral of the first kind, for example:

.

At any point of the integration interval the subintegral function has a discontinuity. In this case, the integral is called improper integral of the second kind, for example:

at the point .

Consider as an example improper integral of the first kind

. The subintegral function plot on the integration interval is depicted below:

Geometrically, this improper integral is equal to the area under the function plot on the interval . The integral in question is convergent because the specified area is equal to 12 – finite number.

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However, the improper integral can also be divergent, for instance:

The algorithm of calculating the improper integral of the first kind:

First of all, we replace the infinite limit with some parameter, for example and get a definite integral. The obtained integral is calculated by usual approach: we find the indefinite integral and then use the Newton-Leibniz formula. At the final stage, we calculate the limit for and if this limit exists and is finite, then the initial improper integral is convergent, otherwise – divergent.