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Common Integration Formulas

∫x^n dx = x^(n+1)/(n+1) + C

Power Rule (n ≠ -1)

∫e^x dx = e^x + C

Exponential Function

∫sin(x) dx = -cos(x) + C

Trigonometric

∫cos(x) dx = sin(x) + C

Trigonometric

∫1/x dx = ln|x| + C

Logarithmic

∫ln(x) dx = x·ln(x) - x + C

Integration by Parts

What is Integration?

Integration is a fundamental concept in calculus that finds the antiderivative of a function. It is the reverse process of differentiation. If differentiation finds the rate of change, integration finds the original function.

Types of Integration

  • Indefinite Integration: Finds the general antiderivative F(x) + C. The constant C represents all possible vertical shifts.
  • Definite Integration: Calculates the area under a curve between two limits a and b.

Basic Rules

∫k dx = kx + C (k is constant)

∫x^n dx = x^(n+1)/(n+1) + C

∫[f(x) + g(x)] dx = ∫f(x)dx + ∫g(x)dx

Applications of Integration

  • Area under curve: Calculate the area bounded by a function and x-axis
  • Volume: Find volume of solids of revolution
  • Physics: Calculate work, electric potential
  • Probability: Find probability distributions

Integration Techniques

Substitution

For composite functions

Integration by Parts

∫u dv = uv - ∫v du

Solved Examples

Example 1: ∫x² dx

Apply power rule: x^(2+1)/(2+1) + C

Answer: x³/3 + C

Example 2: ∫e^x dx

e^x is its own antiderivative

Answer: e^x + C

Example 3: ∫sin(x) dx

Derivative of -cos(x) is sin(x)

Answer: -cos(x) + C

Integration is essential for Class 12 Mathematics, Physics, and Engineering

Advanced Integration Techniques

Trigonometric Substitution

For expressions with √(a²-x²), √(a²+x²), √(x²-a²)

Partial Fractions

Decompose rational functions into simpler fractions

Double Integration

∬f(x,y) dA for area and volume calculations

Integration by Parts (Tabular)

For products of polynomials and exponentials/trig

Fundamental Theorem of Calculus

The Fundamental Theorem links differentiation and integration:

∫ₐᵇ f(x) dx = F(b) - F(a)

Where F(x) is the antiderivative of f(x)

Real-World Applications

  • Engineering: Calculate center of mass, moment of inertia
  • Physics: Find displacement from velocity, work from force
  • Economics: Consumer and producer surplus
  • Statistics: Probability density functions
  • Medicine: Drug concentration over time

Master these techniques to excel in JEE, boards, and engineering entrance exams