1. Introduction to Integration:

Integration is the reverse process of differentiation, and it helps in finding the area under curves. There are two types of integrals: indefinite and definite integrals. Indefinite integrals provide a general form of antiderivatives, while definite integrals compute the exact area between curves and the x-axis.

2. Methods of Integration:

a. Integration by Substitution:

This method simplifies integrals by substituting part of the integrand. The formula is:

If ∫f(g(x))g'(x) dx = ∫f(u) du, where u = g(x)

Example:

Evaluate ∫(2x)/(x² + 1) dx.

Solution: Let u = x² + 1, then du = 2x dx. The integral becomes ∫1/u du = ln|u| + C = ln|x² + 1| + C.

b. Integration by Parts:

Used for integrals involving products of functions, based on: ∫u dv = uv - ∫v du.

READ ALSO: Class 12 Mathematics Notes: Integration as an Inverse Process of Differentiation

Example:

Evaluate ∫x e^x dx.

Solution: Let u = x, dv = e^x dx. Then du = dx, and v = e^x.

∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C.

c. Integration by Partial Fractions:

Used to decompose rational functions into simpler fractions.

Example:

Evaluate ∫1/(x² - x - 2) dx.

Solution: Factor denominator: x² - x - 2 = (x - 2)(x + 1).

Partial fraction decomposition: 1/(x² - x - 2) = A/(x - 2) + B/(x + 1). Solve for A and B.

d. Integration of Trigonometric Functions:

Useful formulas for integrating trigonometric functions include:

∫sin²x dx = (x/2) - (sin(2x)/4) + C, ∫cos²x dx = (x/2) + (sin(2x)/4) + C.

Example:

Evaluate ∫sin²x dx.

3. Definite Integrals:

Definite integrals calculate the area under a curve within limits [a, b].

Example:

Evaluate ∫[0,1] (x² + 1) dx.

Solution: ∫[0,1] (x² + 1) dx = [(x³/3) + x] from 0 to 1 = (1/3 + 1) - (0 + 0) = 4/3.

4. Properties of Definite Integrals:

Important properties:

1. ∫[a,b] f(x) dx = -∫[b,a] f(x) dx

2. ∫[a,a] f(x) dx = 0

3. ∫[a,b] [f(x) ± g(x)] dx = ∫[a,b] f(x) dx ± ∫[a,b] g(x) dx

5. Practice Problems:

1. ∫(3x + 2)/(x² + 4) dx

2. ∫x ln(x) dx

3. ∫sec²x dx

4. ∫1/(x² + 4x + 5) dx

5. ∫[1,2] (x³ - x) dx

Conclusion:

Integration is a fundamental concept in calculus, playing a vital role in various fields such as physics, engineering, and economics. Mastering different methods of integration—such as substitution, parts, partial fractions, and trigonometric integrals—provides a comprehensive toolkit for solving complex problems.

By practicing both definite and indefinite integrals, students gain a deeper understanding of mathematical principles and their practical applications. Remember, consistent practice is key to excelling in this topic.

Table of Contents:

1. Introduction to Integration

2. Methods of Integration

  a. Integration by Substitution

  b. Integration by Parts

  c. Integration by Partial Fractions

  d. Integration of Trigonometric Functions

3. Definite Integrals

4. Properties of Definite Integrals

5. Practice Problems