Repeated Integration by Parts

Repeated integration by parts is an extension of the integration by parts technique in calculus. It is used when you have a product of functions that requires integration multiple times to find the antiderivative. The process involves applying the integration by parts formula iteratively until you reach a point where the resulting integral is easy to evaluate or has a known form.

When applying this formula repeatedly, you would start with an integral that involves a product of two functions, and then apply integration by parts to break it down into simpler integrals. You would then continue this process on the resulting integrals until you reach a point where further applications are unnecessary or where the integrals become manageable.

Here’s a step-by-step example of how repeated integration by parts works:

1. Start with an integral of a product of two functions: ∫ u dv.
2. Apply the integration by parts formula to get: uv – ∫ v du.
3. If the new integral obtained on the right-hand side still involves a product of functions, apply integration by parts again to break it down further.
4. Continue this process until you obtain a simpler integral that can be easily evaluated or one that matches a known integral form.

Tabular Integration by Parts

Tabular integration, also known as the tabular method or the method of tabular integration, is an alternative technique for evaluating integrals that involve repeated application of integration by parts. This method is particularly useful when dealing with integrals where the product of functions can be integrated multiple times to reach a simple result.

The tabular method organizes the repeated integration by parts process into a table, making it easier to keep track of the terms and simplify the integral efficiently. Here’s how the tabular method works:

1. Begin by writing down the functions involved in the integral in two columns: one for the function to differentiate (u) and another for the function to integrate (dv).
   • Start with the function to integrate (dv) on the left column and the function to differentiate (u) on the right column.
2. Continue differentiating the function in the u column until you reach zero or a constant. At each step, integrate the function in the dv column until you reach a point where further integration is not necessary.
3. Multiply the terms diagonally and alternate the signs (+ and -) for each term. Sum up these products to find the result of the integration.

Here’s an example to illustrate the tabular integration method:

Let’s evaluate the integral ∫x sin(x) dx.

• Step 1: Create a table with two columns for u (function to differentiate) and dv (function to integrate):

• Step 2: Differentiate the function in the u column and integrate the function in the dv column:

• Step 3: Multiply the terms diagonally and alternate the signs:

(x)(-cos(x)) – (1)(-sin(x)) + (0)(cos(x)) = -x cos(x) + sin(x)

So, the result of the integral ∫x sin(x) dx is -xcos(x) + sin(x).

The tabular integration method is especially useful when dealing with integrals that involve functions that repeat upon differentiation or integration, allowing for a systematic and organized approach to finding the antiderivative.

Integration by Parts: Integration by parts is a technique used in calculus to find the integral of the product of two functions. It’s essentially a reversal of the product rule for differentiation.

Integrating a function is not always easy sometimes we have to integrate a function that is the multiple of two or more functions in this case if we have to find the integration we have to use integration by part concept, which uses two products of two functions and tells us how to find their integration.

Now let’s learn about Integration by parts, its formula, derivation, and others in detail in this article.