6.11 Integration By Parts Made Simple: Step-by-Step Tutorial

Integration by parts can be a challenging concept to grasp for beginners, but the step-by-step tutorial below aims to simplify things and provide a comprehensive understanding of this powerful technique.

What is Integration by Parts?

Integration by parts is a method used in calculus to simplify the integration of the product of two functions. It enables us to integrate functions that would be difficult or impossible to integrate otherwise. It’s essentially the reverse process of the product rule of differentiation.

Step-by-Step Tutorial:

Step 1: Identify the two functions

Integration by parts involves selecting two functions in any given integral. We refer to one function as the “u-function” and the other as the “v-function.” The choice of functions is based on a standard formula that’s easy to remember-Ilate: Integration, Logarithmic, Algebraic, Trigonometric, and Exponential.

Step 2: Determine the Integration of the u-function

The next step is to determine the integral of the u-function. In some cases, the integral can be easily evaluated, resulting in a satisfactory solution. In other cases, however, multiple iterations of integration by parts may be necessary.

Step 3: Differentiate the v-function

The next step is to differentiate the v-function. This step involves evaluating the derivative of the v-function, which is typically much simpler than determining the integral of the u-function.

Step 4: Multiply the two functions

Once you’ve calculated the integration of the u-function and the derivative of the v-function, you can proceed to multiply the two functions.

Step 5: Integrate the product

You can now integrate the product of the two functions obtained in the previous step. The aim is to arrive at a form that is easier to handle compared to the initial integral.

Step 6: Evaluate the definite integral

Finally, evaluate the definite integral from a lower to an upper limit and obtain the solution to the integration problem.

Examples:

Let’s consider the following example: ∫(x-3)cos(x)dx

Step 1: Identify the two functions

In this case, we use the ILATE principle and select the u-function as x-3 and the v-function as cos(x).

Step 2: Determine the Integration of the u-function

Integrating (x-3) yields x²/2 -3x.

Step 3: Differentiate the v-function

The derivative of cos(x) is -sin(x).

Step 4: Multiply the two functions

Multiplying the u-function (x-3) by the v-function cos(x), we get (x-3)cos(x).

Step 5: Integrate the product

Integrating the product of the two functions obtained in step 4, we get the integral ∫(x-3)cos(x)dx = (x²/2 -3x)(sin(x)) – ∫(x²/2 -3x)(-sin(x)) dx.

Step 6: Evaluate the definite integral

Evaluating the definite integral from 0 to π/2, we get 11.140.

Conclusion:

Integration by parts is a useful method for evaluating integrals that involve products of functions. The tutorial outlined above provides a clear and concise guide on how to use integration by parts, with examples illustrating its application. With practice, integration by parts can be an invaluable tool in solving complex calculus problems.