Exercise 8.2 | Q#1-6 | Evaluate the integral using integration by parts |

Introduction

This tutorial will guide you through the process of evaluating integrals using integration by parts, as demonstrated in Exercise 8.2 from the Mathematics specialist YouTube channel. Integration by parts is a powerful technique used to solve integrals that may not be straightforward. This method is particularly useful for integrals involving products of functions.

Step 1: Identify the Functions

• Determine the two functions from your integral that will be used in the integration by parts formula.
• Typically, you will want to choose:
  • u (a function to differentiate)
  • dv (a function to integrate)

Practical Tip:

• A common mnemonic to remember which function to choose is LIATE:
  • Logarithmic
  • Inverse trigonometric
  • Algebraic
  • Trigonometric
  • Exponential
• Choose u from the function that appears first in this list.

Step 2: Apply the Integration by Parts Formula

• The integration by parts formula is:

  [ \int u , dv = uv - \int v , du ]

• Calculate:

  • The derivative of u to find du.
  • The integral of dv to find v.

Example:

If you have the integral of ( x e^x ):

• Let ( u = x ) (thus, ( du = dx ))
• Let ( dv = e^x , dx ) (thus, ( v = e^x ))

Step 3: Substitute and Simplify

• Substitute u, v, du, and dv into the integration by parts formula:

  [ \int x e^x , dx = x e^x - \int e^x , dx ]

• Solve the remaining integral on the right side.

Common Pitfall:

• Ensure you correctly apply the integral of ( dv ). For example, the integral of ( e^x ) is ( e^x ), not just ( e ).

Step 4: Final Integration

• Integrate the simplified integral. Continuing from the previous example:

  [ \int e^x , dx = e^x ]

• Combine your results:

  [ \int x e^x , dx = x e^x - e^x + C ]

Conclusion

In this tutorial, you learned how to evaluate integrals using integration by parts. The key steps involve identifying functions, applying the integration by parts formula, substituting and simplifying, and finally integrating the remaining expression. Practice this method with various functions to solidify your understanding. As a next step, try evaluating different integrals where integration by parts is applicable to enhance your skills.