JEE · Mathematics

Integration

Find indefinite integrals by recognising antiderivative structure and selecting substitution, integration by parts, partial fractions or trigonometric identities while preserving domains and the constant of integration.

Subject: Mathematics
Syllabus unit: Integral Calculus

Explicit in both JEE Main 2026 Unit 8 and JEE Advanced 2026 Integral Calculus
The constant +C is mandatory on every indefinite integral
No invented weightage, question counts or trend percentages

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In short

An indefinite integral represents a family of antiderivatives. If F'(x)=f(x) on an interval, then ∫f(x)dx=F(x)+C.

Method selection matters: substitution requires a corresponding differential transformation, integration by parts reverses the product rule, and partial fractions requires a valid rational-function decomposition.

Syllabus mapping

Syllabus mapping
Unit	Topics
Integral Calculus	Integration as antiderivative, Fundamental integrals, Substitution, Integration by parts, Partial fractions, Trigonometric identities in integration

Unit
Integral Calculus
Topics
Integration as antiderivative, Fundamental integrals, Substitution, Integration by parts, Partial fractions, Trigonometric identities in integration

What this chapter contains and why it matters

What this chapter contains and why it matters
Question	Direct answer
What is the chapter about?	Finding indefinite integrals by recognising antiderivative structure and selecting substitution, integration by parts, partial fractions, or trigonometric identities.
What is the central method choice?	Match the integrand to a standard derivative, or select substitution, parts, or partial fractions based on its structure.
Where do most mistakes begin?	Omitting +C, calling a relabelling 'substitution' without transforming the differential, and using partial fractions before polynomial division.
What should come before Integration?	Differentiation and Trigonometry for antiderivative recognition and identity simplification.
What comes after it?	Definite Integrals apply fixed limits and the Fundamental Theorem to these same antiderivative techniques.

Question
What is the chapter about?
Direct answer
Finding indefinite integrals by recognising antiderivative structure and selecting substitution, integration by parts, partial fractions, or trigonometric identities.
Question
What is the central method choice?
Direct answer
Match the integrand to a standard derivative, or select substitution, parts, or partial fractions based on its structure.
Question
Where do most mistakes begin?
Direct answer
Omitting +C, calling a relabelling 'substitution' without transforming the differential, and using partial fractions before polynomial division.
Question
What should come before Integration?
Direct answer
Differentiation and Trigonometry for antiderivative recognition and identity simplification.
Question
What comes after it?
Direct answer
Definite Integrals apply fixed limits and the Fundamental Theorem to these same antiderivative techniques.

Official JEE syllabus documents define content scope. They do not publish chapter weightage, so none is asserted here.

Official JEE syllabus mapping for Integration

Verified against the current JEE Main 2026 syllabus and JEE Advanced 2026 syllabus.

Official JEE syllabus mapping for Integration
Concept group	JEE Main 2026	JEE Advanced 2026	Preparation note
Integration as antiderivative	Explicitly listed in Unit 8, with fundamental integrals of algebraic, trigonometric, exponential and logarithmic functions.	Explicitly listed as integration as inverse differentiation with indefinite integrals of standard functions.	Both papers require the same antiderivative base; Main additionally prints simple rational/radical integral families.
Substitution, parts, partial fractions	Explicitly listed in Unit 8.	Explicitly listed.	Practise method selection, not memorised outcomes.

Concept group
Integration as antiderivative
JEE Main 2026
Explicitly listed in Unit 8, with fundamental integrals of algebraic, trigonometric, exponential and logarithmic functions.
JEE Advanced 2026
Explicitly listed as integration as inverse differentiation with indefinite integrals of standard functions.
Preparation note
Both papers require the same antiderivative base; Main additionally prints simple rational/radical integral families.
Concept group
Substitution, parts, partial fractions
JEE Main 2026
Explicitly listed in Unit 8.
JEE Advanced 2026
Explicitly listed.
Preparation note
Practise method selection, not memorised outcomes.

Where a Main-printed simple-integral family produces a closed form not shown on this page, that closed form requires Mathematics SME sign-off before publication.

Before this chapter

Prerequisites: what you should know before Integration

Prerequisites: what you should know before Integration
Prerequisite	You are ready if you can…	If not, repair this first
Differentiation rules	Differentiate a proposed antiderivative to check it.	Revise chain and product rules in Differentiation.
Algebraic factorisation	Factor a denominator before partial fractions.	Revise polynomial factorisation.
Trigonometric identities	Simplify a trig integrand before integrating.	Revise identities in Trigonometry.
Logarithm domains	State the domain of ln\|x\|.	Revise logarithm properties.

Prerequisite
Differentiation rules
You are ready if you can…
Differentiate a proposed antiderivative to check it.
If not, repair this first
Revise chain and product rules in Differentiation.
Prerequisite
Algebraic factorisation
You are ready if you can…
Factor a denominator before partial fractions.
If not, repair this first
Revise polynomial factorisation.
Prerequisite
Trigonometric identities
You are ready if you can…
Simplify a trig integrand before integrating.
If not, repair this first
Revise identities in Trigonometry.
Prerequisite
Logarithm domains
You are ready if you can…
State the domain of ln|x|.
If not, repair this first
Revise logarithm properties.

This is a readiness check, not a weightage or scoring-priority list.

Concepts in this chapter

1. Identify whether the integrand matches a standard derivative

Start from the integrand structure, not from a memorised list of tricks.

Identify whether the integrand already matches a standard antiderivative before searching for a technique.

2. Check the real domain and choose a working interval

Choose an interval avoiding poles where required.

Check the integrand's real domain and choose a working interval that avoids poles or excluded points before integrating.

3. Look for an inner function with its derivative

Substitution requires the differential, not just a change of symbol.

If u=g(x) and the integrand contains the corresponding g'(x)dx, transform the entire differential consistently, not merely relabel a variable.

4. Use integration by parts only when it simplifies the remaining integral

For products, test parts only if the resulting integral is simpler.

For products of unlike functions, consider integration by parts, ∫u dv=uv-∫v du, only if it produces a simpler remaining integral.

5. Divide first, then decompose

Perform polynomial division on an improper rational function before partial fractions.

For rational functions, perform polynomial division if the integrand is improper, then factor the denominator and decompose into partial fractions.

6. Add +C and differentiate to verify

Every indefinite integral answer ends with +C and a derivative check.

Add the constant of integration, then differentiate the final expression on the stated interval to confirm it reproduces the original integrand.

Method selector: match integrand structure to technique

Match the integrand signal to the correct first method before integrating.

Method selector: match integrand structure to technique
Integrand signal	Method
Exact derivative pattern	Standard antiderivative
g'(x)·F(g(x))	Substitution
Product of unlike functions	Test integration by parts
Proper rational function	Partial fractions if denominator factorises suitably
Improper rational function	Polynomial division first
Trig powers/products	Identities if they simplify to standard forms
Quadratic/radical official Main family	Complete square or substitution only after domain/form recognition

Integrand signal
Exact derivative pattern
Method
Standard antiderivative
Integrand signal
g'(x)·F(g(x))
Method
Substitution
Integrand signal
Product of unlike functions
Method
Test integration by parts
Integrand signal
Proper rational function
Method
Partial fractions if denominator factorises suitably
Integrand signal
Improper rational function
Method
Polynomial division first
Integrand signal
Trig powers/products
Method
Identities if they simplify to standard forms
Integrand signal
Quadratic/radical official Main family
Method
Complete square or substitution only after domain/form recognition

Worked reasoning: substitution matched to the differential

Evaluate ∫2x cos(x²) dx

Known: the inner function x² has derivative 2x.

Method choice: substitution exactly matches the differential.

Let u=x², so du=2x dx. Then ∫2x cos(x²)dx = ∫cosu du = sinu + C. Back-substitute: sin(x²)+C.

Validity check: d/dx[sin(x²)+C] = cos(x²)·2x, which exactly reproduces the integrand.

Standard integrals and their conditions

Standard integrals and their conditions
Record	Expression	Conditions
Power	∫x^n dx = x^(n+1)/(n+1) + C	n ≠ -1; real-domain restrictions apply if n is not an integer
Reciprocal	∫dx/x = ln\|x\| + C	x ≠ 0; work on an interval not crossing 0
Exponential	∫e^x dx = e^x + C	all real x
General exponential	∫a^x dx = a^x/ln a + C	a > 0, a ≠ 1
Sine	∫sinx dx = -cosx + C	radians
Cosine	∫cosx dx = sinx + C	radians
Secant squared	∫sec²x dx = tanx + C	interval avoiding cosx = 0
Cosecant squared	∫csc²x dx = -cotx + C	interval avoiding sinx = 0
Inverse tangent form	∫dx/(1+x²) = tan⁻¹x + C	all real x
Inverse sine form	∫dx/√(1-x²) = sin⁻¹x + C	open interval \|x\| < 1
Scaled inverse tangent	∫dx/(a²+x²) = (1/a)tan⁻¹(x/a) + C	a > 0
Scaled inverse sine	∫dx/√(a²-x²) = sin⁻¹(x/a) + C	a > 0, \|x\| < a
Substitution	If u=g(x) and integrand contains g'(x)dx, transform the differential consistently	g differentiable on the working interval; transformed integral valid
Parts	∫u dv = uv - ∫v du	u differentiable and v an antiderivative of dv on the interval
Partial fractions	Decompose a rational function after denominator factorisation	Denominator nonzero on working interval; divide first if numerator degree ≥ denominator degree

Record
Power
Expression
∫x^n dx = x^(n+1)/(n+1) + C
Conditions
n ≠ -1; real-domain restrictions apply if n is not an integer
Record
Reciprocal
Expression
∫dx/x = ln|x| + C
Conditions
x ≠ 0; work on an interval not crossing 0
Record
Exponential
Expression
∫e^x dx = e^x + C
Conditions
all real x
Record
General exponential
Expression
∫a^x dx = a^x/ln a + C
Conditions
a > 0, a ≠ 1
Record
Sine
Expression
∫sinx dx = -cosx + C
Conditions
radians
Record
Cosine
Expression
∫cosx dx = sinx + C
Conditions
radians
Record
Secant squared
Expression
∫sec²x dx = tanx + C
Conditions
interval avoiding cosx = 0
Record
Cosecant squared
Expression
∫csc²x dx = -cotx + C
Conditions
interval avoiding sinx = 0
Record
Inverse tangent form
Expression
∫dx/(1+x²) = tan⁻¹x + C
Conditions
all real x
Record
Inverse sine form
Expression
∫dx/√(1-x²) = sin⁻¹x + C
Conditions
open interval |x| < 1
Record
Scaled inverse tangent
Expression
∫dx/(a²+x²) = (1/a)tan⁻¹(x/a) + C
Conditions
a > 0
Record
Scaled inverse sine
Expression
∫dx/√(a²-x²) = sin⁻¹(x/a) + C
Conditions
a > 0, |x| < a
Record
Substitution
Expression
If u=g(x) and integrand contains g'(x)dx, transform the differential consistently
Conditions
g differentiable on the working interval; transformed integral valid
Record
Parts
Expression
∫u dv = uv - ∫v du
Conditions
u differentiable and v an antiderivative of dv on the interval
Record
Partial fractions
Expression
Decompose a rational function after denominator factorisation
Conditions
Denominator nonzero on working interval; divide first if numerator degree ≥ denominator degree

The Main syllabus additionally prints families involving quadratic denominators, square-root quadratic forms and linear numerators over those forms; closed forms beyond this table require Mathematics SME sign-off before indexation.

Common mistakes and what they actually indicate

Omitting +C on an indefinite integral.
Execution error
Why it happens
An indefinite integral represents an entire family of antiderivatives, not a single function.
How it is corrected
Always append +C to a final indefinite-integral answer.
Calling a change of symbol 'substitution' without transforming the differential.
Decision / selection error
Why it happens
Substitution requires the differential g'(x)dx to be present and transformed, not just a relabelled variable.
How it is corrected
Confirm du=g'(x)dx is genuinely present before calling the method substitution.
Using lnx instead of ln|x| for ∫dx/x on a general real interval.
Knowledge gap
Why it happens
lnx is undefined for x<0, but ln|x| is a valid antiderivative on both x>0 and x<0 branches.
How it is corrected
Write ln|x|+C and state the working interval when it excludes x=0.
Using partial fractions before polynomial division of an improper rational function.
Execution error
Why it happens
Partial-fraction decomposition applies to a proper rational function; an improper one must be divided first.
How it is corrected
Check numerator versus denominator degree and divide first when the fraction is improper.
Using a formula across poles as though one antiderivative expression defines one connected interval.
Needs review
Why it happens
Antiderivative formulas are valid on a connected interval avoiding singularities; carrying one expression across a pole is invalid.
How it is corrected
State the interval explicitly and treat each side of a singularity separately.
Importing definite-integral properties onto an indefinite integral.
Knowledge gap
Why it happens
Properties such as limit reversal or interval addition apply to definite integrals with fixed limits, not to antiderivative families.
How it is corrected
Keep indefinite-integral reasoning separate from definite-integral property use.

FAQ

Integration — questions

Straight answers about how Rank Sarthi fits into serious exam preparation.

Because differentiation removes constants, infinitely many antiderivatives differ only by a constant; +C represents that entire family.

Sources and provenance

Method classification for official Main-printed simple integral families is taught here; closed forms beyond the listed standard integrals require Mathematics SME sign-off before indexation. Use only verified official-paper provenance for worked examples. Do not publish counts or trends.

Official JEE (Main) 2026 syllabus (PDF) — National Testing Agency (NTA) · official-pdf · checked 8 September 2026
JEE (Advanced) 2026 Syllabus (PDF) — JEE (Advanced) 2026, organising institute IIT Roorkee · official-pdf · checked 27 August 2026
NCERT Class XII Mathematics Exemplar: Integrals (PDF) — NCERT · textbook · checked 8 September 2026
NCERT Exemplar Problems index — NCERT · textbook · checked 8 September 2026

Contributor requirements for this page

Written by: unassigned. Ideal author type: Calculus/Mathematical Analysis educator.
Academically reviewed by: unassigned. Reviewer specialisation: standard integrals, domain intervals, substitution validity, integration by parts, partial fractions. Minimum qualification: postgraduate degree in Mathematics or Applied Mathematics.
Review scope: all standard integrals, domain intervals, substitution validity, parts identity, partial fractions, official Main printed family treatment and +C.