A Guide to Using the Integration by Parts Formula in Calculus with an Example

Integration by parts formula

The integration by parts formula is a technique used in calculus to evaluate integrals of products of functions

The integration by parts formula is a technique used in calculus to evaluate integrals of products of functions. It is derived from the product rule for differentiation. The formula is typically written as:

∫ u dv = uv – ∫ v du

where u and v are functions.

To use the integration by parts formula, you need to choose which function to assign as “u” and which one as “dv”. The goal is to differentiate u and integrate dv in order to simplify the integral on the right-hand side.

The formula states that the integral of the product of two functions u and dv is equal to the product of u and v, minus the integral of v and du. This process is repeated iteratively until a simpler integral is obtained that can be easily evaluated.

Let’s go through an example to illustrate how to use the integration by parts formula.

Example:
Evaluate the integral ∫ x ln(x) dx.

Solution:
In this example, we assign “u” to be ln(x) and “dv” to be x dx.
Differentiating u gives du = (1/x) dx.
Integrating dv gives v = (1/2)x^2.

Using the integration by parts formula:

∫ x ln(x) dx = uv – ∫ v du

= ln(x) * (1/2)x^2 – ∫ (1/2)x^2 * (1/x) dx

= (1/2) x^2 ln(x) – (1/2) ∫ x dx

= (1/2) x^2 ln(x) – (1/4) x^2 + C

So, the solution to the integral ∫ x ln(x) dx is (1/2) x^2 ln(x) – (1/4) x^2 + C, where C is the constant of integration.

It’s important to note that there can be cases where you need to apply the integration by parts formula multiple times to simplify the integral further. Additionally, choosing “u” and “dv” requires some strategic selection based on the properties of the functions involved.

I hope this explanation helps you understand the integration by parts formula better. If you have any further questions, please let me know.

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