Lecture Notes on Integration

Introduction

• Integration and Differentiation relationship
  • Differentiation: ds/dt (velocity)
  • Integration: method to obtain ds (displacement)

Need for Integration

• When displacement is given and we need to find velocity.
• Integration is the opposite process of differentiation.

Integration with Graphs

• Finding the area under the graph of Y versus X.
• Calculation of area by making thin strips:
  • Area = Y1 * dx (first strip)
  • Area = Y2 * dx (second strip)
• Adding all strips' areas = summation (Σ) Y dx

Fundamental Formulas of Integration

1. Integration of x^n:

   [ \int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) ]

2. Integration of 1/x:

   [ \int \frac{1}{x} , dx = \log |x| + C ]

3. Integration of e^x:

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

4. Integration of dx:

   [ \int dx = x + C ]

Rules of Integration

• Rule: If a number is multiplied, take it out:
  • For example, [ \int k , f(x) , dx = k \int f(x) , dx ]

Methods of Integration

• Integration by Parts:
  • [ \int u , dv = u , v - \int v , du ]

Special Integrations

• Integration of Trigonometric Functions:
  • [ \int , sin(x) , dx = -cos(x) + C ]
  • [ \int , cos(x) , dx = sin(x) + C ]

Definite and Indefinite Integration

• Definite Integration: Limits are set, such as:
  • [ \int_{a}^{b} f(x) , dx = F(b) - F(a) ]_

Examples

1. Integration of sin(2x):
   • [ \int sin(2x) , dx = -\frac{1}{2} \cos(2x) + C ]
2. Integration of e^{4x}:
   • [ \int e^{4x} , dx = \frac{1}{4} e^{4x} + C ]

Conclusion

• Integration is an important mathematical process, used in physics and other fields.
• It is essential to remember various forms and formulas of integration.

Study Material

• Note down all formulas and practice regularly.
• Solve additional examples and problems.