CBSE Class 12 Mathematics Integration 2026: Every Formula, Method, and Trick to Score Full Marks in Calculus

Integration is the most challenging and most marks-intensive topic in Class 12 Mathematics. Chapters 7 (Integrals) and 8 (Application of Integrals) together account for 15-18 marks in the CBSE board paper — more than any other two chapters combined. Students who master integration have a substantial advantage not just in boards but in JEE Main where calculus questions appear across multiple sections.

This guide presents every integration formula, method, and problem-solving technique you need for Class 12 board exams and beyond, with worked examples and common mistake alerts.

Why Integration is Different from Differentiation

Differentiation has a clear algorithm — apply rules (chain, product, quotient) and you get a unique answer. Integration is the reverse process, and unlike differentiation, there is no single universal algorithm. Integration requires recognising which technique applies to which type of integral, and this pattern recognition is learned through practice, not memorisation alone.

Every integration formula is simply the reverse of a differentiation formula. d/dx(sin x) = cos x, therefore ∫cos x dx = sin x + C. This reverse-engineering logic, once internalised, makes the large formula table much easier to retain.

The Complete List of Standard Integration Formulas

Algebraic integrals:

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
• ∫(1/x) dx = ln|x| + C
• ∫1 dx = x + C
• ∫k dx = kx + C

Trigonometric integrals:

• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫sec²x dx = tan x + C
• ∫cosec²x dx = -cot x + C
• ∫sec x tan x dx = sec x + C
• ∫cosec x cot x dx = -cosec x + C
• ∫tan x dx = ln|sec x| + C
• ∫cot x dx = ln|sin x| + C
• ∫sec x dx = ln|sec x + tan x| + C
• ∫cosec x dx = ln|cosec x – cot x| + C

Exponential and logarithmic integrals:

• ∫eˣ dx = eˣ + C
• ∫aˣ dx = aˣ/ln a + C
• ∫ln x dx = x ln x – x + C (derived by integration by parts)

Inverse trigonometric integrals (most important for boards):

• ∫1/√(1-x²) dx = sin⁻¹x + C
• ∫-1/√(1-x²) dx = cos⁻¹x + C
• ∫1/(1+x²) dx = tan⁻¹x + C
• ∫1/(a²+x²) dx = (1/a) tan⁻¹(x/a) + C
• ∫1/√(a²-x²) dx = sin⁻¹(x/a) + C
• ∫1/√(x²-a²) dx = ln|x + √(x²-a²)| + C
• ∫1/√(x²+a²) dx = ln|x + √(x²+a²)| + C
• ∫1/(x²-a²) dx = (1/2a) ln|(x-a)/(x+a)| + C
• ∫1/(a²-x²) dx = (1/2a) ln|(a+x)/(a-x)| + C

Method 1: Integration by Substitution

Substitution is used when the integrand contains a function and its derivative. The substitution reduces the complex integral to a standard form.

Rule: If the integral is ∫f(g(x)) g'(x) dx, substitute u = g(x), du = g'(x) dx. The integral becomes ∫f(u) du.

When to use substitution: When you can spot a function inside another function, and the derivative of the inner function is also present (or can be made to appear by adjusting constants).

Example: ∫2x(x²+1)⁵ dx. Let u = x²+1, then du = 2x dx. Integral becomes ∫u⁵ du = u⁶/6 + C = (x²+1)⁶/6 + C.

Trigonometric substitution: For integrals involving √(a²-x²), use x = a sin θ. For √(a²+x²), use x = a tan θ. For √(x²-a²), use x = a sec θ. These substitutions convert radical expressions into trigonometric forms that are easier to integrate.

Method 2: Integration by Parts

Integration by parts is the reverse of the product rule of differentiation. Formula: ∫u v dx = u∫v dx – ∫(du/dx × ∫v dx) dx.

The ILATE rule helps choose which function to call u (the first function): Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. Select u as whichever function comes first in ILATE. The other function is v.

Example: ∫x sin x dx. By ILATE: u = x (Algebraic), v = sin x (Trigonometric). Then: ∫x sin x dx = x(-cos x) – ∫(1)(-cos x) dx = -x cos x + sin x + C.

Special case — ∫eˣ[f(x) + f'(x)] dx: This equals eˣ f(x) + C. Recognising this pattern saves enormous time for a class of problems that appears repeatedly in board exams. Example: ∫eˣ(sin x + cos x) dx = eˣ sin x + C (since d/dx(sin x) = cos x).

Method 3: Partial Fractions

Partial fractions is used for integrals of rational functions (polynomial/polynomial) where the degree of numerator < degree of denominator. The fraction is decomposed into simpler fractions that are easier to integrate individually.

Forms of partial fractions:

• (px+q)/[(x-a)(x-b)] = A/(x-a) + B/(x-b)
• (px+q)/[(x-a)²] = A/(x-a) + B/(x-a)²
• (px²+qx+r)/[(x-a)(x²+bx+c)] = A/(x-a) + (Bx+C)/(x²+bx+c)

To find constants A, B, C: multiply both sides by the denominator and substitute suitable values of x (typically the roots of each factor). This gives a system of equations to solve for A, B, C.

Common board question type: ∫1/[(x-1)(x+2)] dx. Partial fractions: 1/[(x-1)(x+2)] = A/(x-1) + B/(x+2). Setting x=1: A=1/3. Setting x=-2: B=-1/3. Integral = (1/3)ln|x-1| – (1/3)ln|x+2| + C = (1/3)ln|(x-1)/(x+2)| + C.

Definite Integrals: Properties That Save Time

Definite integrals ∫[a to b] f(x) dx have important properties that frequently reduce complex integrals to simple ones:

Property 1: ∫[a to b] f(x) dx = -∫[b to a] f(x) dx (swapping limits changes sign).

Property 2: ∫[a to b] f(x) dx = ∫[a to b] f(a+b-x) dx (king’s property). This is the most useful property for evaluating integrals like ∫[0 to π] x sin x/(1+cos²x) dx — substitute x → π-x and add.

Property 3: ∫[0 to 2a] f(x) dx = 2∫[0 to a] f(x) dx if f(2a-x) = f(x), or = 0 if f(2a-x) = -f(x).

Property 4 (even/odd functions): If f(-x) = f(x) (even): ∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx. If f(-x) = -f(x) (odd): ∫[-a to a] f(x) dx = 0. This is enormously useful for definite integrals with symmetric limits.

Chapter 8: Application of Integrals — Area Under Curves

Chapter 8 applications use definite integrals to find areas bounded by curves and lines. Board questions typically ask for the area between a parabola and a line, between two curves, or between a circle/ellipse and the axes.

Area formula: Area bounded by y = f(x), x-axis, and x = a, x = b is ∫[a to b] f(x) dx. If the curve goes below the x-axis in some interval, split the integral at the x-crossing points and take the absolute value of each part.

Area between two curves: Area = ∫[a to b] [f(x) – g(x)] dx, where f(x) is the upper curve and g(x) is the lower curve. Find intersection points by setting f(x) = g(x) — these are the limits a and b.

Standard area results to memorise:

• Area of circle x²+y²=r² is πr² (verify by integration: 4∫[0 to r]√(r²-x²) dx)
• Area enclosed by parabola y²=4ax and the line x=a is (8a²/3)
• Area of ellipse x²/a² + y²/b² = 1 is πab

For board exams, always draw a rough sketch of the region whose area you are finding. Label the intersection points, identify which curve is on top, and set up the integral before computing. Even a rough diagram earns method marks and helps you avoid sign errors.

Common Mistakes to Avoid

Forgetting the constant of integration C in indefinite integrals — every indefinite integral must end with “+ C”. Errors in integration by parts: wrong choice of u and v by not applying ILATE correctly. Not checking if numerator degree is less than denominator degree before applying partial fractions (if degree of numerator is equal or higher, perform polynomial long division first). Sign errors in trigonometric integrals: ∫sin x dx = -cos x (negative sign) while ∫cos x dx = sin x (positive). These sign errors cause 1-mark losses across multiple questions.

Practice Strategy for Integration Mastery

Week 1: Master all standard formulas by deriving each one from differentiation. Week 2: Practise substitution only (50 problems). Week 3: Integration by parts (30 problems). Week 4: Partial fractions (25 problems). Week 5-6: Mixed problems and definite integrals with properties. Final 2 weeks: Past 5 years board papers, full papers under time conditions.

Integration is a skill that requires practice, not just reading. Students who solve 150+ integration problems before boards almost invariably find the board paper straightforward. Those who only read solutions without practicing independently struggle even with standard questions under time pressure.