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Integral Calculus
Integral Calculus
Integral Calculus – JEE Mains Mathematics
by Zenith LMS
Updated April 2025
1. Integral as an Anti-Derivative
Integration is the reverse operation of differentiation, i.e., the anti-derivative of a function.
If F'(x) = f(x), then ∫f(x) dx = F(x) + C, where C is the constant of integration.
Integration is used to find the area under curves, and solve problems involving rates of change.
2. Fundamental Integrals
Some basic integrals for algebraic, trigonometric, exponential, and logarithmic functions:
∫x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1
∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫e^x dx = e^x + C
∫1/x dx = ln|x| + C
3. Integration Techniques
3.1. Integration by Substitution
If a function is complicated, substitution is used to simplify the integral.
If u = g(x), then ∫f(g(x)) * g'(x) dx = ∫f(u) du
Example: ∫x * cos(x²) dx → let u = x², then du = 2x dx.
3.2. Integration by Parts
Integration by parts is based on the product rule of differentiation:
∫u dv = uv - ∫v du
It is useful when the integral involves a product of two functions.
Example: ∫x * e^x dx → u = x, dv = e^x dx, then du = dx, v = e^x.
3.3. Integration by Partial Fractions
Used to integrate rational functions by expressing them as a sum of simpler fractions.
Example: ∫(1/(x² - 1)) dx can be decomposed into partial fractions.
The technique involves expressing the denominator as a product of linear factors and solving for constants.
3.4. Integration Using Trigonometric Identities
Some integrals can be simplified using trigonometric identities, such as:
sin²(x) = (1 - cos(2x))/2
cos²(x) = (1 + cos(2x))/2
Using these, integrals like ∫sin²(x) dx or ∫cos²(x) dx can be simplified and solved.
4. The Fundamental Theorem of Calculus
It establishes the connection between differentiation and integration.
The first part states that if f(x) is continuous on [a, b], then the function F(x) = ∫[a, x] f(t) dt is differentiable, and its derivative is f(x).
The second part states that if F(x) is the antiderivative of f(x), then ∫[a, b] f(x) dx = F(b) - F(a).
5. Properties of Definite Integrals
If f(x) is continuous on [a, b], then:
∫[a, b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
∫[a, b] f(x) dx = -∫[b, a] f(x) dx (reverse limits).
∫[a, b] [f(x) ± g(x)] dx = ∫[a, b] f(x) dx ± ∫[a, b] g(x) dx.
∫[a, b] c f(x) dx = c ∫[a, b] f(x) dx (constant factor rule).
6. Evaluation of Definite Integrals
To evaluate a definite integral, find the antiderivative of the function and then substitute the limits of integration.
Example: ∫[0, 1] (2x) dx = [x²] from 0 to 1 = 1² - 0² = 1.
7. Determining Areas of Regions Bounded by Curves
Definite integrals can be used to find areas between curves.
If a curve is defined by y = f(x) and bounded by x = a and x = b, the area between the curve and the x-axis is given by:
Area = ∫[a, b] f(x) dx
For regions bounded by two curves y = f(x) and y = g(x), the area between the curves is:
Area = ∫[a, b] [f(x) - g(x)] dx
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JEE Mains Physics
Units and Measurements
Kinematics
Laws of Motion
Work, Energy and Power
Rotational Motion
Gravitation
Properties of Solids and Liquids
Thermodynamics
Kinetic Theory of Gases
Oscillations and Waves
Electrostatics
Current Electricity
Magnetic Effects of Current and Magnetism
Electromagnetic Induction and Alternating Currents
Electromagnetic Waves
Optics
Dual Nature of Matter and Radiation
Atoms and Nuclei
Electronic Devices
Experimental Skills
JEE Mains Chemistry
Some Basic Concepts in Chemistry
Atomic Structure
Chemical Bonding and Molecular Structure
Chemical Thermodynamics
Solutions
Equilibrium
Redox Reactions and Electrochemistry
Chemical Kinetics
Classification of Elements and Periodicity
p-Block Elements
d-Block and f-Block Elements
Coordination Compounds
Purification and Characterisation of Organic Compounds
Some Basic Principles of Organic Chemistry
Hydrocarbons
Organic Compounds Containing Halogens
Organic Compounds Containing Oxygen
Organic Compounds Containing Nitrogen
Biomolecules
Practical Chemistry Principles
JEE Mains Mathematics
Integral Calculus
Application of Derivatives
Continuity and Differentiability
Sets, Relations, and Functions
Complex Numbers Quadratics
Matrices and Determinants
Permutations and Combinations
Binomial Theorem and Its Simple Applications
Sequence and Series
Limit, Continuity, and Differentiability
Differential Equations
Coordinate Geometry
Three Dimensional Geometry
Vector Algebra
Statistics and Probability
Trigonometry