04_Ex 3.4

Exercise Questions

Questions	Question Links
Q1. Evaluate the following integral by parts…	3.4 Q-1
Q2. Evaluate the following integrals…	3.4 Q-2
Q3. Show that…	3.4 Q-3
Q4. Evaluate the following integrals…	3.4 Q-4
Q5. Evaluate the following integrals…	3.4 Q-5

Overview

This exercise covers the method of Integration by Parts (IBP), a key technique used for integrating products of functions. By applying the product rule in reverse, IBP helps evaluate integrals that are not straightforward. Understanding this method and related techniques, such as the ILATE rule, is essential for handling complex integrals in calculus.

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Key Concepts

• Integration by Parts (IBP): A method derived from the product rule for differentiation. It is expressed as:

$$\int u , dv = uv - \int v , du$$

• $u$: A function that is easily differentiable.

• $dv$: A function that is easily integrable.

• ILATE Rule: A mnemonic used to help choose $u$ in integration by parts:

  • I: Inverse functions (e.g., $\sin^{-1}x$, $\tan^{-1}x$)
  • L: Logarithmic functions (e.g., $\ln x$)
  • A: Algebraic functions (e.g., $x^2$, $x^3$)
  • T: Trigonometric functions (e.g., $\sin x$, $\cos x$)
  • E: Exponential functions (e.g., $e^x$)

• Trigonometric and Exponential Integration: Applying specific formulas to simplify integrals of trigonometric and exponential functions.

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Important Formulas

• Integration by Parts:

$$\int u , dv = uv - \int v , du$$

• Trigonometric Integrals:

  • $\int \sin^2 x , dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$
  • $\int \cos^2 x , dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$

• Exponential Integral:

$$\int e^x , dx = e^x + C$$

• Special Integrals:

  • $\int \sqrt{a^2 - x^2} , dx = \frac{a^2}{2} \sin^{-1} \left(\frac{x}{a}\right) + \frac{x}{2} \sqrt{a^2 - x^2} + C$
  • $\int x \sin^2 x , dx = \frac{x^2}{4} - \frac{x \sin 2x}{4} + \frac{1}{8} \cos 2x + C$

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Tips and Tricks

• ILATE Rule: Use the ILATE mnemonic to help decide which function to assign to $u$ in the IBP formula. Choose the function that appears first in the ILATE list.
• Identify Standard Forms: Many integrals, especially those involving trigonometric or exponential functions, have standard forms. Memorizing these can save time during integration.
• Reapply Integration by Parts: For complex integrals, you may need to apply IBP more than once. Be strategic about your choices of $u$ and $dv$.

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Summary

This exercise focuses on Integration by Parts, a method used to integrate products of functions. The key to success is choosing the appropriate functions for $u$ and $dv$ using the ILATE Rule and applying the formula:

$$\int u , dv = uv - \int v , du$$

Familiarity with special integral forms and practicing the method will help you tackle a wide variety of integrals efficiently.

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Reference

By Sir Shahzad Sair:

By Great Science Academy:

Reference provided in Questions as videos are divided