Understanding the Integration by Parts Formula | A Guide to Evaluating Integrals of Product Functions

Integration by parts formula

The integration by parts formula is a method used to evaluate the integral of a product of two functions, known as the “integrand

The integration by parts formula is a method used to evaluate the integral of a product of two functions, known as the “integrand.”

The formula is derived from the product rule of differentiation, and it can be stated as follows:

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

In this formula, u(x) and v(x) are two functions differentiable on some interval, and u'(x) and v'(x) represent their respective derivatives.

To use the integration by parts formula, you need to select u(x) and v'(x) appropriately. The choice of u(x) is generally based on a priority order of functions in decreasing “niceness” or “simplicity.” This priority order is known as the “LIATE” rule:

L – Logarithmic functions
I – Inverse trigonometric functions
A – Algebraic functions (polynomials)
T – Trigonometric functions
E – Exponential functions

The idea is to choose u(x) as the function that is closer to the beginning of the list, and v'(x) as the derivative of the function closer to the end of the list. This typically ensures that the integral in the formula becomes simpler or leads to a known result.

Once you have chosen u(x) and v'(x), you differentiate u(x) to find u'(x), and integrate v'(x) to find v(x). Then, you can plug these values into the integration by parts formula and evaluate the resulting integrals.

It is often necessary to apply the integration by parts formula several times or in combination with other integration techniques to evaluate more complex integrals. With practice and experience, you will develop a better intuition for selecting suitable u(x) and v'(x) functions and navigating through the integrals using this formula.

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