Successive Integration
Successive Integration - Related Videos
02:17
Ecological Succession
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2026
Ecological succession is influenced by the processes of facilitation, inhibition, and toleration. Facilitation occurs when early successional species create more favorable ecological conditions for subsequent species, such as enhanced nutrient, water, or light availability. In contrast, inhibition happens when early successional species create unfavorable ecological conditions for potential successive species, such as limiting resource availability. In some cases, later successional species...
05:05
The Successive Alleys Test of Anxiety in Mice and Rats
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Cited by 49 •
2013
The plus-maze measures anxiety-like behaviour in rodents. There are two opposite closed and two opposite open arms; anxious rodents avoid the open arms. The central area is neither completely open nor closed, so time spent here is ambiguous and difficult to interpret. Here a modification of the plus-maze protocol eliminating this area is described.
12:24
Improving the Success Rate of Protein Crystallization by Random Microseed Matrix Screening
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Cited by 20 •
2013
Here we describe a general method for random microseed matrix screening. This technique is shown to significantly increase the success rate of protein crystallization screening experiments, reduce the need for optimization, and provide a reliable supply of crystals for data collection and ligand-soaking experiments.
06:49
Reduced Itraconazole Concentration and Durations Are Successful in Treating Batrachochytrium dendrobatidis Infection in Amphibians
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Cited by 18 •
2014
Batrachochytrium dendrobatidis infection in amphibians can be eliminated using a reduced concentration (0.0025%) and shorter duration (five minute baths for six days) of itraconazole than is generally used. Less mortality and fewer negative side effects were observed using this modified treatment protocol.
01:26
Integration by Parts: Indefinite Integrals
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2026
Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
FAQs
Each indefinite integration introduces its own arbitrary constant, so the constants from earlier stages cannot be collapsed into a single bookkeeping step. After one antiderivative is obtained, integrating it again adds another independent constant to the next expression. Tracking these terms separately preserves the full family of functions whose successive derivatives match the original function.
Definite successive integration replaces newly introduced arbitrary constants with specified limits of integration. Repeating the operation therefore produces nested integral expressions, where each stage accumulates the result from the preceding stage over its stated interval. This form is useful when the calculation requires a determined value or relationship rather than the unrestricted family produced by indefinite antiderivatives.
The method reverses a chain of differentiations one level at a time. If derivative information is given for a higher-order relationship, integrating once reconstructs the preceding level, and integrating again continues toward the original function. At every stage, the new expression becomes the input for the next reversal, creating a systematic route through higher-order differential equations.
Start by integrating the given function once, then treat the resulting expression as the integrand for the next stage. For an indefinite calculation, append a fresh constant at every integration and keep those terms distinct. For a definite calculation, apply the specified limits at each stage and retain the resulting nested structure. This sequence prevents stages from being conflated.
Successive integration is especially relevant when a problem supplies a function's derivative information at multiple levels or asks for repeated integral evaluation. It can reconstruct functions by reversing successive differentiations and organize multi-stage accumulation calculations. In differential-equation work, the approach provides a direct framework for moving from higher-order derivative relationships toward corresponding antiderivative expressions.
Each integration stage converts the expression from the preceding stage into another accumulated quantity, extending the connection between a function and the rates represented by its derivatives. Repeating this process builds several levels of accumulation rather than stopping after one antiderivative. That perspective helps organize calculations involving repeated integrals and functions reconstructed from derivative information.