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∫ f(x) +g(x) dx = ∫ f(x) dx + ∫ g(x) dx.

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Q: What is the integral of two functions of x added together with respect to x?
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What are some methods to approximate the value of an integral when it cannot be calculated directly?

There are two main methods. One is the trapezium method. Here the integral ("area under the curve") is split into a large number of trapeziums with their parallel sides being vertical. The areas of these trapeziums are calculated and added together to approximate the integral. The greater the number of trapeziums the more accurate the answer but also the more effort is required. Another approach, which can sometimes work, is to find two integrable functions such that one of them is greater than the function to be integrated and the other is less. If the required function can be tightly sandwiched between two such functions then the integral of the function will be between the integrals of the two integrable functions. Even if you cannot find two such functions that fulfil this requirement over the whole domain, provided you can find pairs of functions that do so over all but a finite number of points is good enough. Actually, I believe it works even if the number of gaps are countably infinite (cardinality Aleph null), but I am not sure of that.


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