CBSE Class 12 Applied Maths Integration Full Revision 2026

CBSE Class 12 Applied Maths Integration Full Revision 2026: CBSE Class 12 students preparing for the Applied Maths subject can get a full revision schedule for the Integration topic.CBSE Class 12 Applied Maths Integration Full Revision 2026 includes the complete revision of the important concepts. It can help the students understand the crucial integration-related topics relevant to exam preparation and question practice. 

Definition of Integration

Integration is the anti-derivative of a function. It is the reverse process of differentiation and is used to find the original function when its derivative is known.

Integration of Basic Powers of x

The standard integration formulas for powers of x are: ∫ 1 dx = x + C, ∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) + C, where n ≠ −1.

Special Case

For n = −1, the formula changes: ∫ (1/x) dx = log|x| + C. This happens because the general power rule does not apply when n = −1.

Integration of Exponential Functions

Important exponential integration formulas are: ∫ aˣ dx = aˣ/log a + C and ∫ eˣ dx = eˣ + C.

Constant Factor Rule

If a constant is multiplied by a function, it remains outside the integral: k ∫ f(x) dx = kF(x) + C, where k is a constant.

Methods of Integration

1. Direct Integration

Direct integration involves applying standard formulas directly. Example: ∫ x³ dx = x⁴/4 + C.

2. Integration by Separating Terms

If the function is a sum or difference, integrate each term separately. Example: ∫ (x² + 3x + 5) dx = ∫ x² dx + ∫ 3x dx + ∫ 5 dx.

3. Integration by Substitution

Substitution is used when a complicated expression can be replaced with a simpler variable. Steps include choosing a substitution variable, replacing the expression with the new variable, converting the integral completely, integrating, and substituting back. Example: ∫ (log x)⁹ (1/x) dx. Let t = log x, then dt = (1/x) dx. The integral becomes ∫ t⁹ dt.

4. Integration by Parts

This method is used when the integrand is a product of two functions. The formula is: ∫ u dv = uv − ∫ v du.

Choosing u

Use the LIATE rule: Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, Exponential.

Example

∫ x eˣ dx. Let u = x and dv = eˣ dx, then du = dx and v = eˣ. Applying the formula gives ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.

5. Integration by Partial Fractions

This method is used for rational functions (ratio of polynomials). Steps include factoring the denominator, decomposing into simpler fractions, solving for constants, and integrating each term separately. Example form: 1/(x² − a²) = A/(x − a) + B/(x + a).

6. Special Tricks in Integration

Some integrals require algebraic techniques such as rationalising denominators, using conjugates, completing the square, and applying trigonometric identities.

Detailed Examples and Problem Solving

Integration of Exponential Expressions

Expressions like 2ˣ, 3ˣ, and 2ˣ/3ˣ can be simplified using exponent rules before integration. Example: 2ˣ/3ˣ = (2/3)ˣ and then integrate normally.

Linear Substitution Rule

When a variable is replaced with a linear expression (ax + b), divide the result by the coefficient a. Example: ∫ f(ax + b) dx. Let t = ax + b. Then ∫ f(ax + b) dx = (1/a) ∫ f(t) dt.

Integration of Composite Functions

Sometimes expressions contain roots and powers together. Example: ∫ √(x + 1) dx. These can be simplified using substitution.

Integration of Polynomial and Exponential Products

Example: ∫ x² eˣ dx. This is solved using integration by parts repeatedly.

Partial Fraction Decomposition

For rational functions P(x)/Q(x) where the degree of P(x) is less than Q(x), the fraction is decomposed into simpler fractions. Example: 1/(x² − 1). Factor the denominator as (x − 1)(x + 1) and express it as A/(x − 1) + B/(x + 1).

Special Integrals and Important Formulas

Some standard integrals include: ∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C and ∫ dx/(x² − a²) = (1/2a) log|(x − a)/(x + a)| + C. These formulas are commonly used in integration problems.

Completing the Square

Quadratic expressions such as x² + 4x + 5 can be rewritten as (x + 2)² + 1. It helps apply inverse trigonometric formulas.

Important Notes About the Constant of Integration

Every indefinite integral must include + C, where C is the constant of integration. Example: ∫ x² dx = x³/3 + C. Even when multiple integrations occur, the constant must be included.