Integrals

Integrals are a fundamental concept in calculus, representing the accumulation of quantities and the area under curves. They are the inverse operation of differentiation and are widely used in mathematics, physics, engineering, and other fields.

1. Introduction to Integrals

• Integral calculus deals with finding the function when its derivative is known.
• The process of finding integrals is called integration.
• Integrals are used to calculate areas, volumes, central points, and many useful things.

2. Types of Integrals

• Indefinite Integrals: Integrals without limits, representing a family of functions. Written as ∫f(x)dx.
• Definite Integrals: Integrals with upper and lower limits, representing a specific value (area under the curve). Written as ∫abf(x)dx.

3. Basic Integration Formulas

• ∫xndx = (xn+1)/(n+1) + C, where n ≠ -1
• ∫exdx = ex + C
• ∫1/x dx = ln|x| + C
• ∫cos(x)dx = sin(x) + C
• ∫sin(x)dx = -cos(x) + C

4. Properties of Definite Integrals

• ∫aaf(x)dx = 0
• ∫abf(x)dx = -∫baf(x)dx
• ∫ab[f(x) + g(x)]dx = ∫abf(x)dx + ∫abg(x)dx

5. Methods of Integration

• Substitution Method
• Integration by Parts
• Partial Fractions
• By Using Trigonometric Identities

6. Applications of Integrals

• Finding area under curves
• Calculating volumes of solids of revolution
• Finding displacement from velocity
• Solving problems in physics and engineering

7. Example Problems

1. Indefinite Integral:
   ∫(3x2 + 2x + 1)dx
   Solution: = x3 + x2 + x + C
2. Definite Integral:
   ∫02 x dx
   Solution: = [x2/2]02 = (4/2) - (0/2) = 2
3. Area under the curve:
   Find the area under y = x from x = 0 to x = 3.
   Solution: ∫03 x dx = [x2/2]03 = (9/2) - 0 = 4.5

8. Practice Exercises

• Find ∫(2x + 5)dx
• Evaluate ∫14 x2 dx
• Find the area under y = 2x from x = 0 to x = 5
• Integrate sin(x) + cos(x) with respect to x

9. Summary

• Integration is the reverse process of differentiation.
• Indefinite integrals give a family of functions; definite integrals give a number (area, volume, etc.).
• Integrals have wide applications in mathematics and science.