Solving Integrals of Product Functions: A Comprehensive Guide

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In summary, finding the integral for any integral of the type f(x)g(x)dx can be done using integration by parts. This method involves treating either g(x) or f(x) as a derivative, similar to g'(x), and only one of the functions needs to be integrable while the other needs to be differentiable. However, for more complex integrals such as (x^n)(e^(x^(m+p)))dx, a case-by-case approach may be necessary. Alternatively, for contour integrals, the residues of f(z)g(z) can be obtained using the Laurent series for f(z) and g(z) in certain cases. Overall, integration by parts is a versatile method that can be applied to
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satxer
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How do I find the integral for any integral of the type: f(x)g(x)dx

I've looked every where, and the closest I've seen is integration by parts which is apparently for the integral of f(x)g'(x)dx, which, to my inexpert eyes, is a completely different integral than the first one

Also, this is probably harder and I think I'm less likely to get an answer, but how about the integral of (x^n)(e^(x^(m+p)))dx
 
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  • #2
It's similar to a convolution, which has some nice properties, but other than that they usually need to be dealt with on a case by case basis. On the other hand if you wish to perform a contour integral you can get the residues of f(z)g(z) with the Laurent series for f(z) and g(z) in certain cases.
 
  • #3
Nono, integration by parts is what you're looking for I'm pretty sure. The only difference is that you would treat either g(x) or f(x) as a derivative, like g'(x). To use integration by parts only one of the functions you're given has to be integrable, where the other needs to be differentiable. Integration by parts is pretty malleable. (Sp?)

Hope this helps.
 

1. What is the definition of the integral of products?

The integral of products refers to the mathematical operation of finding the area under the curve of a product of two functions. It is a way to measure the accumulated change of a product over an interval. It is denoted by the symbol ∫ and is commonly used in calculus and other branches of mathematics.

2. How is the integral of products calculated?

The integral of products is calculated using the fundamental theorem of calculus. This involves finding the antiderivative of the product function and evaluating it at the upper and lower limits of the integral. The difference between these two values gives the value of the integral.

3. What are the applications of the integral of products?

The integral of products has many practical applications in fields such as physics, engineering, and economics. It can be used to calculate the work done by a variable force, the displacement of an object under varying velocity, and the area under a demand curve to determine the total revenue or profit.

4. Are there any special cases for calculating the integral of products?

Yes, there are some special cases for calculating the integral of products. One of them is the case of a constant factor, where the integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function. Another special case is when the two functions being multiplied are the same, resulting in the integral of the square of a function.

5. What are some techniques for evaluating the integral of products?

Some common techniques for evaluating the integral of products include substitution, integration by parts, and trigonometric substitution. These techniques allow for the conversion of a complex product function into a simpler form that can be easily integrated. It is important to choose the most suitable technique based on the given product function.

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