Integration by U-substitution

Integration by U-substitution method
When the integrand is formed by a product (or a division, which we can treat like a product) it's recommended the use of the method known as integration by u-substitution, that consists in applying the following formula:
Even though it's a simple formula, it has to be applied correctly. Let's see a few tips on how to apply it well:
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1. Select u and dv correctly: as a rule, we will call u all powers and logarithms; and dv exponentials, fractions and trigonometric functions (circular functions).
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2. Don't change our minds about the selection: Sometimes we need to apply the method more than once for the same integral. When this happens, we need to call u the result of du from the first integral we applied the method to.
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3. Cyclic integrals: Sometimes, after applying integration by u-substitution twice we have to isolate the very integral from the equality we've obtained in order to resolve it. An example of this is example 3.
Examples
Example 1:
Notes: it's important to choose
x = u, so dx = du
because by doing so we're reducing the monomials degree (from 1 to 0). If we choose
x = dv, so v = x^2/2
we increase the degree (from 1 to 2) and we complicate the integral more because the exponential factor remains the same.
Example 2:
In this integral we don't have an explicit product of functions, but we don't know what the logarithms primitive function is, so we differentiate it, that way u = ln(x).
Example 3 (cyclic integral):
In this example it doesn't matter which factors are u and dv, because when integrating and differentiating e -x we obtain –e -x and when integrating and differentiating cos(x) we get ±sin(x). This is a cyclical integral in which we have to apply integration by u-substitution twice (with the same choices so we don't go backwards) and we have to isolate the integral from the mathematical expression we obtain.
More examples: U-substitution method.
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