CBSE Class 12 Maths Formulas – Complete Chapter Wise Formula Sheet

Maths Formulas for Class 12 Chapter wise

Maths Formulas for Class 12 PDF Free Download is a helpful resource for students preparing for board exams and competitive tests. In Class 12 Mathematics, students study important topics like calculus, integration, differentiation, matrices, determinants, vectors, probability, and three-dimensional geometry. Remembering all formulas can be difficult, so having a clear and simple formula sheet makes revision faster and easier.

This maths formulas for class 12 pdf free download includes all important formulas chapter-wise in one place. It helps students quickly revise key concepts, solve numerical problems easily, and improve their exam performance with regular practice.

Chapter 1: Relations and Functions

What It Is

Relations and Functions form the foundation of higher mathematics. A relation maps elements of one set to another; a function is a special relation where every input has exactly one output.

Formulas and Concepts

Formula / Concept	Expression	Explanation	Variables	Exam Use Case
Number of Relations	2^(n×m)	Total relations from set A (n elements) to set B (m elements)	n, m = cardinality of sets	Finding total possible relations
Number of Functions	m^n	Functions from A (n elements) to B (m elements)	n = |A|, m = |B|	Counting onto/into functions
Bijective Function	f: A→B is one-one and onto	Both injective and surjective	—	Proving a function is bijective
Composition of Functions	(gof)(x) = g(f(x))	Apply f first, then g	f, g = functions	Composite function problems
Inverse of a Function	f⁻¹(f(x)) = x	Exists only for bijective functions	—	Finding f⁻¹

Facts:

• A function is one-one (injective) if f(a) = f(b) ⟹ a = b
• A function is onto (surjective) if range = codomain
• Binary operation * on set A: * : A × A → A

Also Check – CBSE Class 12 Chemistry Formulas | Complete Class 12 Physics Formulas

Chapter 2: Inverse Trigonometric Functions

What It Is

These formulas define the inverse of trig functions with restricted domains so they remain functions.

Domain and Range Table

Function	Domain	Range (Principal Value Branch)
sin⁻¹(x)	[-1, 1]	[-π/2, π/2]
cos⁻¹(x)	[-1, 1]	[0, π]
tan⁻¹(x)	ℝ	(-π/2, π/2)
cot⁻¹(x)	ℝ	(0, π)
sec⁻¹(x)	ℝ \ (-1, 1)	[0, π] \ {π/2}
cosec⁻¹(x)	ℝ \ (-1, 1)	[-π/2, π/2] \ {0}

Important Identities

Identity	Formula	When to Use
Sine inverse property	sin⁻¹(sin x) = x	x ∈ [-π/2, π/2]
Complementary angles	sin⁻¹(x) + cos⁻¹(x) = π/2	Simplifying expressions
Tan + Cot identity	tan⁻¹(x) + cot⁻¹(x) = π/2	All x ∈ ℝ
Sec + Cosec identity	sec⁻¹(x) + cosec⁻¹(x) = π/2	|x| ≥ 1
Negative argument	sin⁻¹(-x) = -sin⁻¹(x)	Odd function property
cos⁻¹ negative	cos⁻¹(-x) = π – cos⁻¹(x)	Even-like property
tan⁻¹ addition	tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1-xy))	xy < 1
tan⁻¹ subtraction	tan⁻¹x – tan⁻¹y = tan⁻¹((x-y)/(1+xy))	xy > -1
2tan⁻¹ formula	2tan⁻¹x = sin⁻¹(2x/(1+x²))	x ∈ [-1, 1]
2tan⁻¹ cosine form	2tan⁻¹x = cos⁻¹((1-x²)/(1+x²))	x ≥ 0

Memory Trick: “SIN + COS = π/2, TAN + COT = π/2, SEC + COSEC = π/2” — All complementary pairs sum to π/2!

Chapter 3: Matrices

Formulas

Formula	Expression	Note
Order of matrix	m × n	m = rows, n = columns
Addition condition	A + B is defined if A and B have same order	—
Scalar multiplication	(kA)ᵢⱼ = k·Aᵢⱼ	Each element multiplied by k
Matrix multiplication	(AB)ᵢⱼ = Σ Aᵢₖ·Bₖⱼ	Columns of A = Rows of B
Transpose of AB	(AB)ᵀ = BᵀAᵀ	Order reverses
Symmetric matrix	A = Aᵀ	aᵢⱼ = aⱼᵢ
Skew-symmetric	A = -Aᵀ	aᵢⱼ = -aⱼᵢ, diagonal = 0
Any matrix as sum	A = ½(A + Aᵀ) + ½(A – Aᵀ)	Symmetric + Skew-symmetric
Identity matrix	AI = IA = A	I has 1s on diagonal
Null matrix	A + 0 = A	Zero matrix

Special Matrix Types

• Diagonal matrix: Non-diagonal elements = 0
• Scalar matrix: Diagonal matrix with equal diagonal elements
• Idempotent matrix: A² = A

Chapter 4: Determinants

Formulas

Formula	Expression	Usage
2×2 Determinant	|A| = ad – bc for [[a,b],[c,d]]	Basic determinant
Area of triangle	½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|	Coordinate geometry
Adjoint	adj(A) = Cᵀ (transpose of cofactor matrix)	Matrix inverse
Inverse of matrix	A⁻¹ = adj(A)/|A|	|A| ≠ 0
Singular matrix	|A| = 0	Inverse does not exist
Cramer’s Rule (x)	x = D₁/D	Solving linear equations
Cramer’s Rule (y)	y = D₂/D	Solving linear equations
Cramer’s Rule (z)	z = D₃/D	3-variable system
Property: Row ops	|kA| = kⁿ|A| for n×n matrix	Scalar multiplication
Transpose property	|Aᵀ| = |A|	Useful in proofs

Cofactor and Minor

• Minor Mᵢⱼ = determinant after deleting row i and column j
• Cofactor Cᵢⱼ = (-1)^(i+j) × Mᵢⱼ

Chapter 5: Continuity and Differentiability

Continuity Condition

A function f(x) is continuous at x = a if: lim(x→a) f(x) = f(a) (i.e., LHL = RHL = f(a))

Differentiation Formulas

Function	Derivative	Formula
xⁿ	nxⁿ⁻¹	Power Rule
eˣ	eˣ	Exponential
aˣ	aˣ · ln(a)	General exponential
ln(x)	1/x	Natural log
log_a(x)	1/(x·ln a)	Log base a
sin(x)	cos(x)	Trig
cos(x)	-sin(x)	Trig
tan(x)	sec²(x)	Trig
cot(x)	-cosec²(x)	Trig
sec(x)	sec(x)·tan(x)	Trig
cosec(x)	-cosec(x)·cot(x)	Trig
sin⁻¹(x)	1/√(1-x²)	Inverse trig
cos⁻¹(x)	-1/√(1-x²)	Inverse trig
tan⁻¹(x)	1/(1+x²)	Inverse trig
cot⁻¹(x)	-1/(1+x²)	Inverse trig
sec⁻¹(x)	1/(x·√(x²-1))	Inverse trig
cosec⁻¹(x)	-1/(x·√(x²-1))	Inverse trig

Main Theorems

Theorem	Statement	Use
Chain Rule	dy/dx = (dy/du)·(du/dx)	Composite functions
Product Rule	d/dx(uv) = u·v’ + v·u’	Product of two functions
Quotient Rule	d/dx(u/v) = (v·u’ – u·v’)/v²	Division of functions
Rolle’s Theorem	f'(c) = 0 for some c ∈ (a,b)	f(a)=f(b), f continuous & differentiable
Mean Value Theorem	f'(c) = [f(b)-f(a)]/(b-a)	f continuous on [a,b], diff. on (a,b)
Logarithmic diff.	y = f(x)^g(x) → take log both sides	Complex exponential forms

Chapter 6: Application of Derivatives

Critical Formulas

Concept	Formula	Application
Rate of change	dy/dx represents rate of y w.r.t. x	Speed, growth problems
Slope of tangent	m = f'(x₀) at point (x₀, y₀)	Equation of tangent
Equation of tangent	y – y₀ = f'(x₀)(x – x₀)	Tangent line
Equation of normal	y – y₀ = -1/f'(x₀) · (x – x₀)	Normal line
Increasing function	f'(x) > 0 on (a,b)	Monotonicity
Decreasing function	f'(x) < 0 on (a,b)	Monotonicity
First derivative test	f'(x) changes sign at c → local extremum	Maxima/Minima
Second derivative test	f”(c) < 0 → local max; f”(c) > 0 → local min	Maxima/Minima
Point of inflection	f”(x) = 0	Concavity changes
Approximation	Δy ≈ f'(x)·Δx	Small change formula

Chapter 7: Integrals

Standard Integration Formulas

Integral	Result	Note
∫ xⁿ dx	xⁿ⁺¹/(n+1) + C	n ≠ -1
∫ 1/x dx	ln|x| + C	—
∫ eˣ dx	eˣ + C	—
∫ aˣ dx	aˣ/ln(a) + C	—
∫ sin(x) dx	-cos(x) + C	—
∫ cos(x) dx	sin(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	—
∫ sec²(x) dx	tan(x) + C	—
∫ cosec²(x) dx	-cot(x) + C	—
∫ sec(x)tan(x) dx	sec(x) + C	—
∫ cosec(x)cot(x) dx	-cosec(x) + C	—
∫ 1/√(1-x²) dx	sin⁻¹(x) + C	—
∫ -1/√(1-x²) dx	cos⁻¹(x) + C	—
∫ 1/(1+x²) dx	tan⁻¹(x) + 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	—
∫ 1/(x²+a²) dx	(1/a)·tan⁻¹(x/a) + C	—
∫ 1/√(a²-x²) dx	sin⁻¹(x/a) + C	—
∫ √(a²-x²) dx	(x/2)√(a²-x²) + (a²/2)sin⁻¹(x/a) + C	—

Integration by Parts (ILATE Rule)

Formula: ∫ u·v dx = u·∫v dx − ∫(u’ · ∫v dx) dx

ILATE Order of preference for u: I – Inverse trigonometric L – Logarithmic A – Algebraic T – Trigonometric E – Exponential

Special Integrals

Form	Formula
∫ eˣ[f(x) + f'(x)] dx	eˣ·f(x) + C
∫ √(x²±a²) dx	(x/2)√(x²±a²) ± (a²/2)ln|x+√(x²±a²)| + C

Definite Integral Properties

Property	Formula
Symmetry	∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx
Zero boundary	∫ₐᵃ f(x)dx = 0
Reverse limits	∫ₐᵇ f(x)dx = -∫ᵦₐ f(x)dx
Additive	∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵦ f(x)dx
Even function	∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f(-x)=f(x)
Odd function	∫₋ₐᵃ f(x)dx = 0 if f(-x)=-f(x)
King’s rule	∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx

Chapter 8: Application of Integrals

Area Formulas

Situation	Formula	Notes
Area under curve (x-axis)	A = ∫ₐᵇ |f(x)| dx	From x=a to x=b
Area under curve (y-axis)	A = ∫ₐᵇ |g(y)| dy	g(y) is x as function of y
Area between two curves	A = ∫ₐᵇ [f(x) – g(x)] dx	f(x) ≥ g(x) on [a,b]
Area of circle	A = πr² (verify via ∫)	∫₋ᵣʳ √(r²-x²)dx × 2
Area of ellipse	A = πab	a, b = semi-axes

Chapter 9: Differential Equations

Formulas and Methods

Concept	Formula / Method	Application
Order	Highest derivative present	Classification
Degree	Power of highest derivative	Classification
Variable Separable	f(y)dy = g(x)dx → integrate both sides	Simple separable equations
Homogeneous Equation	Put y = vx, dy/dx = v + x·dv/dx	f(x,y) = f(tx,ty)
Linear DE (standard)	dy/dx + P(x)y = Q(x)	First-order linear
Integrating Factor	IF = e^(∫P dx)	Linear DE solution
Solution of linear DE	y × IF = ∫Q × IF dx + C	General solution
General Solution	Contains arbitrary constant(s)	Full family of solutions
Particular Solution	Constant determined by initial condition	Specific solution

Chapter 10: Vector Algebra

Core Vector Formulas

Formula	Expression	Meaning
Magnitude of vector	|a| = √(a₁²+a₂²+a₃²)	Length of vector
Unit vector	â = a/|a|	Direction, magnitude = 1
Dot product	a·b = |a||b|cosθ	Scalar result
Dot product (components)	a·b = a₁b₁+a₂b₂+a₃b₃	Component form
Angle between vectors	cosθ = (a·b)/(	|a||b|)
Cross product magnitude	|a×b| = |a||b|sinθ	Vector result
Cross product (det form)	a×b = |î ĵ k̂ / a₁ a₂ a₃ / b₁ b₂ b₃|	Component calculation
Area of parallelogram	|a×b|	Two sides a and b
Area of triangle	½|a×b|	Two sides a and b
Scalar triple product	[abc] = a·(b×c)	Volume of parallelepiped
Volume of parallelepiped	|[abc]|	Scalar triple product
Coplanar vectors	[abc] = 0	Test for coplanarity
Projection of a on b	(a·b)/|b|	Scalar projection

Position Vector and Section Formula

Formula	Expression
Midpoint	(a + b)/2
Section formula (internal)	(mb + na)/(m+n)
Section formula (external)	(mb – na)/(m-n)

Chapter 11: Three Dimensional Geometry

This chapter has the highest formula density in Class 12. Master each type carefully.

Direction Cosines and Ratios

Formula	Expression	Use
Direction cosines	l = cosα, m = cosβ, n = cosγ	Angles with x, y, z axes
Fundamental relation	l² + m² + n² = 1	Always true for DCs
DC from DR	l = a/√(a²+b²+c²), similarly m, n	Converting DRs to DCs
DC of line joining (x₁,y₁,z₁) to (x₂,y₂,z₂)	(x₂-x₁)/d, (y₂-y₁)/d, (z₂-z₁)/d	d = distance

Equations of a Line

Form	Equation	Variables
Cartesian (point + DR)	(x-x₁)/a = (y-y₁)/b = (z-z₁)/c	(x₁,y₁,z₁) = point, (a,b,c) = DR
Vector form	r = a + λb	a = position vector, b = direction
Two-point form	(x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁)	Two known points

Equations of a Plane

Form	Equation	Notes
Normal form (vector)	r·n̂ = d	n̂ = unit normal, d = distance from origin
Cartesian normal form	lx + my + nz = d	l,m,n = DCs of normal
Intercept form	x/a + y/b + z/c = 1	Intercepts a, b, c on axes
Through 3 points	Determinant form	Use cofactor expansion
Passing through point, parallel to two vectors	(r–a)·(b×c) = 0	—

Distance Formulas

Formula	Expression	Use
Distance: point to plane	d = |ax₁+by₁+cz₁+d|/√(a²+b²+c²)	Plane ax+by+cz+d=0
Distance: point to line	d = |(b–a)×d|/|d|	Vector form
Distance between parallel planes	d = |d₁-d₂|/√(a²+b²+c²)	Same normal direction
Shortest distance (skew lines)	SD = |(a₂–a₁)·(b₁×b₂)| / |b₁×b₂|	Skew line formula

Angle Formulas

Between	Formula
Two lines	cosθ = |l₁l₂ + m₁m₂ + n₁n₂|
Line and plane	sinθ = |al+bm+cn|/[√(a²+b²+c²)·√(l²+m²+n²)]
Two planes	cosθ = |a₁a₂+b₁b₂+c₁c₂|/[√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)]

Exam Tip: For Chapter 11, always identify whether you’re working with a line or a plane before choosing a formula. Mixing them up is the #1 source of errors.

Chapter 12: Linear Programming

Concepts (No Complex Formulas — Logic-Based)

Concept	Definition
Objective Function	Z = ax + by (to maximize or minimize)
Constraints	Linear inequalities restricting variables
Feasible Region	All points satisfying all constraints
Corner Point Method	Evaluate Z at each vertex of feasible region
Optimal Solution	Max or min value of Z at a corner point
Bounded Region	Closed feasible region
Unbounded Region	Open feasible region — check if max/min exists

Rule: If the feasible region is bounded, the optimal value always exists. If unbounded, verify using the open half-plane test.

Chapter 13: Probability

Core Probability Formulas

Formula	Expression	Use
Basic probability	P(A) = n(A)/n(S)	Classical definition
Complement rule	P(A’) = 1 – P(A)	—
Addition rule	P(A∪B) = P(A)+P(B)-P(A∩B)	Any two events
Mutually exclusive	P(A∪B) = P(A) + P(B)	A∩B = ∅
Conditional probability	P(A|B) = P(A∩B)/P(B)	B already occurred
Multiplication rule	P(A∩B) = P(A)·P(B|A)	Joint probability
Independent events	P(A∩B) = P(A)·P(B)	A and B independent
Bayes’ Theorem	P(Eᵢ|A) = P(Eᵢ)·P(A|Eᵢ) / Σ P(Eⱼ)·P(A|Eⱼ)	Reverse conditional
Total Probability	P(A) = Σ P(Eᵢ)·P(A|Eᵢ)	Exhaustive events

Binomial Distribution

Element	Formula	Meaning
P(X = r)	ⁿCᵣ · pʳ · qⁿ⁻ʳ	r successes in n trials
Mean	np	Expected value
Variance	npq	q = 1-p
Standard Deviation	√(npq)	—

Solved Examples

Example 1: Inverse Trig (Chapter 2)

Problem: Find the value of sin⁻¹(sin 5π/6).

Solution: 5π/6 is NOT in [-π/2, π/2], so we cannot directly apply sin⁻¹(sin x) = x.

sin(5π/6) = sin(π – π/6) = sin(π/6) = 1/2

Therefore, sin⁻¹(sin 5π/6) = sin⁻¹(1/2) = π/6

Example 2: Maxima/Minima

Problem: Find the maximum value of f(x) = 2x³ – 15x² + 36x + 4.

Solution: Step 1: f'(x) = 6x² – 30x + 36 = 6(x² – 5x + 6) = 6(x-2)(x-3)

Step 2: f'(x) = 0 → x = 2 or x = 3

Step 3: f”(x) = 12x – 30

• At x = 2: f”(2) = 24-30 = -6 < 0 → Local Maximum
• At x = 3: f”(3) = 36-30 = 6 > 0 → Local Minimum

Step 4: f(2) = 16 – 60 + 72 + 4 = 32 (Maximum value)

Example 3: Shortest Distance

Problem: Find shortest distance between lines: r = (î+2ĵ+3k̂) + λ(2î+3ĵ+4k̂) and r = (2î+4ĵ+5k̂) + μ(4î+6ĵ+8k̂)

Solution: Notice b₂ = 2b₁ → Lines are parallel (not skew).

For parallel lines: SD = |(a₂-a₁)×b₁| / |b₁|

a₂-a₁ = î+2ĵ+2k̂, b₁ = 2î+3ĵ+4k̂

Cross product = |î ĵ k̂ / 1 2 2 / 2 3 4| = î(8-6) – ĵ(4-4) + k̂(3-4) = 2î – k̂

|Cross product| = √(4+0+1) = √5

|b₁| = √(4+9+16) = √29

SD = √5/√29 = √(5/29) units

Example 4: Probability with Bayes’ Theorem (Chapter 13)

Problem: A bag has 3 red and 5 black balls. A ball is drawn at random. If it’s red, what’s the probability it came from the first draw being red?

Common Mistakes to Avoid

Chapter 2 (Inverse Trig):

• Writing sin⁻¹(sin 5π/6) = 5π/6 — WRONG. Always check if x is in the principal value branch.

Chapter 5 (Differentiation):

• Forgetting the chain rule when differentiating composite functions like sin(x²).

Chapter 7 (Integration):

• Missing the constant of integration C in indefinite integrals.
• Applying the wrong ILATE priority in integration by parts.

Chapter 11 (3D Geometry):

• Confusing Direction Cosines with Direction Ratios.
• Using the wrong formula for skew lines vs parallel lines.

Chapter 13 (Probability):

• Applying Bayes’ Theorem without checking that events form a partition of the sample space.

Memory Tips & Tricks

• ILATE for integration by parts: I nverse, L og, A lgebraic, T rig, E xponential.
• “All Silver Tea Cups” for trig signs in quadrants: All (+), Sin (+), Tan (+), Cos (+).
• For inverse trig pairs: sin⁻¹ + cos⁻¹ = π/2 — remember “they always add to 90°.”
• For 3D: “DCs are always normalized, DRs are not” — DCs satisfy l²+m²+n² = 1.
• Matrices: “Transpose reverses the order of products” — (AB)ᵀ = BᵀAᵀ, not AᵀBᵀ.
• Determinants: Expanding along the row/column with most zeros saves computation time.
• Integration: When you see √(a²-x²), think sin substitution; √(x²-a²), think sec substitution.

Conclusion

Class 12 Maths formulas are not just symbols on a page — they are the tools that transform a difficult problem into a solved one.

This chapter-wise formula sheet covers every essential formula across all 13 chapters: from the elegant symmetry of inverse trigonometric identities, to the power of integration techniques, to the spatial precision of 3D geometry. Each formula here has been tested in board exams for years.

Note for your Exam:

• Relations and Functions: Know bijection, composition, and inverse.
• Calculus (Ch. 5–8): Differentiation rules, integration formulas, and area applications.
• 3D Geometry (Ch. 11): Master direction cosines, line/plane equations, and distance formulas this chapter alone carries 8–12 marks.
• Probability (Ch. 13): Understand conditional probability, Bayes’ Theorem, and binomial distribution deeply.

Revision is not about reading it’s about active recall. Close this article, take a blank sheet, and write down every formula you remember. The gaps you find are exactly what to revise next.

Class 12 Maths Formulas related FAQs

Q. Which chapter has the most formulas in Class 12 Maths?

Chapter 7 (Integrals) and Chapter 11 (Three Dimensional Geometry) have the highest number of formulas. Integrals alone contain 25+ standard results, while 3D Geometry covers lines, planes, distances, and angles comprehensively.

Q. How many formulas are there in Class 12 Maths in total?

There are approximately 150–200 formulas across all 13 chapters of Class 12 Maths. However, for board exams, mastering around 80–100 high-frequency formulas is sufficient for scoring 90+.

Q. What are the most important formulas for Class 12 board exams?

The most exam-critical formulas include integration by parts (Chapter 7), shortest distance between skew lines (Chapter 11), Bayes’ Theorem (Chapter 13), maxima-minima (Chapter 6), and properties of definite integrals (Chapter 7).

Q. How do I memorize Class 12 Maths formulas effectively?

Write formulas by hand daily, practice applying each formula with 2–3 problems, use mnemonics like ILATE, group related formulas together, and revise your formula sheet every 3 days before exams.

Q. Is there a difference between direction cosines and direction ratios?

Yes. Direction cosines (l, m, n) are the actual cosines of angles a line makes with the coordinate axes and always satisfy l²+m²+n²=1. Direction ratios (a, b, c) are any set of numbers proportional to DCs but not necessarily normalized.

Q. What is the formula for the shortest distance between two skew lines?

SD = |(a₂-a₁)·(b₁×b₂)| / |b₁×b₂|, where a₁, a₂ are position vectors of points on the lines and b₁, b₂ are direction vectors. This applies only when lines are skew (non-parallel, non-intersecting).

Q. Can I score 100/100 in Class 12 Maths by just learning formulas?

Knowing formulas is necessary but not sufficient. You also need to understand when and how to apply each formula, practice solving complete problems, and manage exam time. Formulas + practice + concept clarity = 100/100.

Q. What is the King’s rule in definite integrals?

The King’s property states: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx. It’s one of the most powerful tools in Class 12 integration problems, especially when direct integration is difficult but adding f(x) + f(a-x) simplifies to a constant.