Mastering Integration By Parts: A Comprehensive Guide For Evaluating Integrals

Integration by Parts

uv – int(vdu)

Integration by parts is a technique used to evaluate the integral of a product of two functions. Mathematically, it is stated as:

∫u(x)v'(x) dx = u(x)v(x) – ∫v(x)u'(x) dx

Where u(x) and v(x) are two functions and u'(x) and v'(x) are their respective derivatives. It is also known as the “ILATE” rule, where one function takes priority over the other.

To use integration by parts, we choose one expression as the u(x) and the other one as v'(x). To determine which expression should be u(x), a common acronym to follow is “ILATE”. This acronym stands for: inverse trigonometric functions, logarithms, algebraic functions, trigonometric functions, and exponential functions.

Once we determine which function will be the u(x), we take the derivative of it to obtain u'(x). We then integrate v'(x) to obtain v(x). From there, we can plug these values into the integration by parts formula and solve for the original integral.

It is important to note that sometimes we need to use integration by parts multiple times in order to solve the integral completely. The key is to choose the best function to be u(x) to simplify the integral by reducing the derivative of u(x) or integrating v'(x).

Overall, integration by parts can be a powerful tool for evaluating integrals when used correctly.

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