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COCoNuT
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when doing integration, how do you know if you should use u-du substitution or integration by parts if the problem doesn't state it?
U-substitution is a technique used to simplify integrals by substituting a variable with a new expression. Integration by parts is a method used to integrate products of functions by breaking them down into simpler integrals.
U-substitution is typically used when the integral contains a single term, while integration by parts is used when the integral contains products of functions. However, the choice between the two techniques ultimately depends on the complexity of the integral and what yields a simpler result.
The u should be chosen such that its derivative, du, appears in the integral. This can be done by identifying a function within the integral that has a derivative that can be expressed in terms of u.
Yes, it is possible to use both techniques in succession if the integral cannot be easily solved with just one method. In this case, u-substitution is typically used first to simplify the integral, followed by integration by parts to integrate any remaining products of functions.
Yes, there are several other techniques for solving integrals, such as trigonometric substitution, partial fraction decomposition, and the method of undetermined coefficients. These techniques may be useful for specific types of integrals, but u-substitution and integration by parts are the most commonly used methods.