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 - finite number. 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.
The algorithm of calculating the improper integral of the second kind consists in splitting the integration interval into segments in each of which the subintegral function is continuous (discontinuities are only allowed at the ends of the segments). Further, the obtained definite integrals are calculated by applying the corresponding limits when using Newton-Leibniz formula. And if all these limits exist and are finite, then, as before, the integral is convergent, otherwise - divergent. Consider an example:
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