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Integrals

Integrals in Class 12 Mathematics begin with the idea that integration reverses differentiation. If the derivative of a function is known, integration helps recover the original family of functions, with an arbitrary constant in indefinite integrals. The chapter develops several methods for handling different types of integrands: direct standard forms, substitution, trigonometric identities, partial fractions, and integration by parts. Choosing the correct method is often the main exam skill. Definite integrals connect integration with signed area and accumulated change through limits. The Fundamental Theorem of Calculus converts a definite integral into a value found from an antiderivative. Properties of definite integrals are used to simplify difficult-looking integrals, especially those involving symmetry, interval reversal, and transformations such as replacing x by a minus x.

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high importance8 concepts

All Concepts

All 8 concepts of the chapter, in the order the NCERT chapter teaches them.

Key Concepts

Work through the concepts in order. Each one has an explanation, a worked example, a self-check and an exam answer.

All Concepts

All 8 concepts of the chapter, in the order the NCERT chapter teaches them.

8 concepts

medium difficulty

Integration as the Reverse of Differentiation

If F'(x)=f(x), then the indefinite integral of f(x) is written as ∫f(x) dx=F(x)+C, where C is an arbitrary constant.

6 minOpen concept

medium difficulty

Integration by Substitution

Integration by substitution changes a difficult integral into a standard one by putting u=g(x), so that du=g'(x) dx matches part of the integrand.

7 minOpen concept

medium difficulty

Integration Using Trigonometric Identities

Some trigonometric integrals are first simplified using identities, such as sin²x=(1-cos2x)/2 and cos²x=(1+cos2x)/2, before applying standard integration formulas.

6 minOpen concept

medium difficulty

Standard Integrals of Particular Forms

Integrals of particular forms are standard results for expressions such as 1/(x²+a²), 1/(x²-a²), 1/√(a²-x²), and related algebraic forms.

6 minOpen concept

medium difficulty

Integration by Partial Fractions

Integration by partial fractions decomposes a proper rational function into simpler fractions whose integrals are standard, usually logarithmic or inverse trigonometric.

7 minOpen concept

medium difficulty

Integration by Parts

Integration by parts uses the formula ∫u dv=uv-∫v du to integrate a product of two functions, with u chosen so that its derivative becomes simpler.

7 minOpen concept

medium difficulty

Definite Integrals and the Fundamental Theorem of Calculus

If F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx=F(b)-F(a). This result is the evaluation form of the Fundamental Theorem of Calculus.

6 minOpen concept

medium difficulty

Properties of Definite Integrals

Properties of definite integrals transform the limits or integrand to simplify evaluation without first finding a complicated antiderivative.

7 minOpen concept

Exam Intelligence

The traps that cost marks in this chapter, and how its main concepts turn up in the paper.

Common Traps

Omitting +C in indefinite integrals.
Using differentiation rules instead of integration rules for powers.
Missing constant factors during substitution, especially du multipliers.
Applying the wrong standard form because the sign in x²±a² was not checked.
Skipping polynomial division before partial fractions when the rational function is improper.
Using the wrong sign in ∫u dv=uv-∫v du.
Adding +C after evaluating a definite integral.
Reversing definite integral limits without changing the sign.

How CBSE Asks This Chapter

Integration as the Reverse of Differentiation
Usually asked as a direct 1-mark or 2-mark question, sometimes mixed with standard integrals of powers, exponentials, trigonometric functions, and simple algebraic sums.
Integration by Substitution
Appears in short-answer and long-answer questions with algebraic, exponential, logarithmic, and trigonometric composite functions. Definite integral versions may test correct transformation of limits.
Integration Using Trigonometric Identities
Questions may ask direct evaluation, simplification before integration, or comparison of equivalent answers. Long-answer questions may combine identities with substitution.
Standard Integrals of Particular Forms
Often asked as direct standard-form questions or after completing the square. May also appear inside substitution-based integrals where the transformed integral matches a standard form.

Quick Revision

Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.

Integration as the Reverse of Differentiation
Remember
Key rules: ∫xⁿ dx=xⁿ⁺¹/(n+1)+C for n≠-1; ∫kf(x) dx=k∫f(x) dx; ∫[f(x)+g(x)] dx=∫f(x) dx+∫g(x) dx.
Trap to avoid
Students often omit +C in indefinite integrals or divide by the old power incorrectly in ∫xⁿ dx.
Integration by Substitution
Remember
Method condition: if the integral has the form ∫f(g(x))g'(x) dx, put u=g(x), du=g'(x) dx, then integrate ∫f(u) du and substitute back.
Trap to avoid
A common error is choosing u correctly but forgetting to replace dx and the remaining factor, leaving a mixture of u and x.
Integration Using Trigonometric Identities
Remember
Useful identities: sin²x=(1-cos2x)/2, cos²x=(1+cos2x)/2, 2sinxcosx=sin2x, sin²x+cos²x=1, 1+tan²x=sec²x.
Trap to avoid
Students sometimes integrate sin²x as if it were sin x, which is incorrect because powers of trigonometric functions need reduction or another suitable identity.
Standard Integrals of Particular Forms
Remember
Key forms for a>0: ∫dx/(x²+a²)=(1/a)tan⁻¹(x/a)+C; ∫dx/√(a²-x²)=sin⁻¹(x/a)+C; ∫dx/(x²-a²)=(1/(2a))ln|(x-a)/(x+a)|+C; ∫dx/(a²-x²)=(1/(2a))ln|(a+x)/(a-x)|+C.
Trap to avoid
A frequent mistake is confusing x²+a² with a²-x² and using tan inverse when sine inverse is required.
Integration by Partial Fractions
Remember
Method condition: use partial fractions for rational functions P(x)/Q(x), after ensuring degree P < degree Q.
Trap to avoid
Students often skip checking whether the fraction is proper.
Integration by Parts
Remember
∫u dv=uv-∫v du.
Trap to avoid
A common mistake is choosing u poorly, making the remaining integral harder instead of simpler.
Definite Integrals and the Fundamental Theorem of Calculus
Remember
FTC evaluation rule: ∫_a^b f(x) dx=F(b)-F(a), where F'(x)=f(x) and f is integrable on [a,b].
Trap to avoid
Students sometimes add +C in a definite integral or substitute limits in the wrong order as F(a)-F(b).
Properties of Definite Integrals
Remember
Important properties: ∫_a^b f(x)dx=-∫_b^a f(x)dx; ∫_a^b f(x)dx=∫_a^c f(x)dx+∫_c^b f(x)dx; ∫₀^a f(x)dx=∫₀^a f(a-x)dx; ∫_-a^a f(x)dx=0 if f is odd; ∫_-a^a f(x)dx=2∫₀^a f(x)dx if f is even.
Trap to avoid
Students often reverse limits without changing the sign, writing ∫_a^b f(x)dx=∫_b^a f(x)dx, which is false unless the integral is zero.

Concept	Remember	Trap to avoid
Integration as the Reverse of Differentiation	`Key rules: ∫xⁿ dx=xⁿ⁺¹/(n+1)+C for n≠-1; ∫kf(x) dx=k∫f(x) dx; ∫[f(x)+g(x)] dx=∫f(x) dx+∫g(x) dx.`	Students often omit +C in indefinite integrals or divide by the old power incorrectly in ∫xⁿ dx.
Integration by Substitution	`Method condition: if the integral has the form ∫f(g(x))g'(x) dx, put u=g(x), du=g'(x) dx, then integrate ∫f(u) du and substitute back.`	A common error is choosing u correctly but forgetting to replace dx and the remaining factor, leaving a mixture of u and x.
Integration Using Trigonometric Identities	`Useful identities: sin²x=(1-cos2x)/2, cos²x=(1+cos2x)/2, 2sinxcosx=sin2x, sin²x+cos²x=1, 1+tan²x=sec²x.`	Students sometimes integrate sin²x as if it were sin x, which is incorrect because powers of trigonometric functions need reduction or another suitable identity.
Standard Integrals of Particular Forms	`Key forms for a>0: ∫dx/(x²+a²)=(1/a)tan⁻¹(x/a)+C; ∫dx/√(a²-x²)=sin⁻¹(x/a)+C; ∫dx/(x²-a²)=(1/(2a))ln\|(x-a)/(x+a)\|+C; ∫dx/(a²-x²)=(1/(2a))ln\|(a+x)/(a-x)\|+C.`	A frequent mistake is confusing x²+a² with a²-x² and using tan inverse when sine inverse is required.
Integration by Partial Fractions	`Method condition: use partial fractions for rational functions P(x)/Q(x), after ensuring degree P < degree Q.`	Students often skip checking whether the fraction is proper.
Integration by Parts	`∫u dv=uv-∫v du.`	A common mistake is choosing u poorly, making the remaining integral harder instead of simpler.
Definite Integrals and the Fundamental Theorem of Calculus	`FTC evaluation rule: ∫_a^b f(x) dx=F(b)-F(a), where F'(x)=f(x) and f is integrable on [a,b].`	Students sometimes add +C in a definite integral or substitute limits in the wrong order as F(a)-F(b).
Properties of Definite Integrals	`Important properties: ∫_a^b f(x)dx=-∫_b^a f(x)dx; ∫_a^b f(x)dx=∫_a^c f(x)dx+∫_c^b f(x)dx; ∫₀^a f(x)dx=∫₀^a f(a-x)dx; ∫_-a^a f(x)dx=0 if f is odd; ∫_-a^a f(x)dx=2∫₀^a f(x)dx if f is even.`	Students often reverse limits without changing the sign, writing ∫_a^b f(x)dx=∫_b^a f(x)dx, which is false unless the integral is zero.

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