Integration Lecture by C Chamber Jacob

Topic: Solving Integration Problems

Problem A

1. Copy the expression: Start by copying the given expression.
2. Rules for Integration:
   • Check the powers: Identify the power of the variable (e.g., x).
   • Add one to the power: For instance, if x has a power of 2, add 1 (resulting in x^3).
   • Include the variable: When integrating a constant, include the variable (e.g., x).
3. Example:
   • Initial Expression: 6x^2 - 5
   • Add one to the power: 6x^(2+1) - 5x
   • Simplify: 2x^3 - 5x
   • Always add the integration constant (C): 2x^3 - 5x + C

Problem B

1. Copy the initial expression: Write it as given.
2. Handling fractions:
   • Use laws of indices to rearrange fractions involving variables.
   • Convert expressions like 1/x^2 to x^-2.
3. Integrate step-by-step:
   • Add one to the power: x^-2 becomes x^(-2+1) = x^-1.
   • Divide by the new power: Adjust coefficients accordingly.
   • Example Expression: 3x^3/4 - 5x^2/2 + x.
   • Simplify the fractions and include the constant term (C).
4. Final Solution Steps:
   • 3x^4/8 - 5x^2/2 - x^-1 + C
   • Clean up the expression for clarity.

Key Points

• Always add 1 to the power when integrating.
• Always divide by the new power.
• Don’t forget to include the constant of integration (C).
• Simplify expressions for the final answer.
• Use laws of indices to handle fractions and exponents properly.

Conclusion

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