Complete Integration Formulas Guide: Every Formula You Need for Class 12 & Beyond
What is Integration?
Integration is the reverse process of differentiation. It's also called anti-differentiation.
When you integrate a function, you find the original function whose derivative gives you the current function.
Simple analogy: If differentiation tells you the speed at any moment, integration tells you the total distance covered.
Why it matters:
- Calculates areas under curves
- Solves real-world problems in physics and engineering
- Essential for board exams and competitive tests
- Carries significant weightage (15-20 marks in boards)
Basic Integration Formulas
These are the fundamental formulas every student must memorize first.
| Formula Name | Formula | Variables | Notes |
|---|---|---|---|
| Power Rule | ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C | n ≠ -1 | Most frequently used formula |
| Constant Rule | ∫k dx = kx + C | k = constant | Simple linear integration |
| Reciprocal | ∫(1/x) dx = ln|x| + C | x ≠ 0 | Special case when n = -1 |
| Exponential (base e) | ∫eˣ dx = eˣ + C | - | e raised to x remains unchanged |
| Exponential (base a) | ∫aˣ dx = aˣ/ln(a) + C | a > 0, a ≠ 1 | General exponential function |
| Sum Rule | ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx | - | Integration is linear |
| Constant Multiple | ∫k·f(x) dx = k·∫f(x) dx | k = constant | Constants can be taken outside |
Memory Tip:
For power rule, just add 1 to the power and divide by the new power. Always remember "+ C" (constant of integration)!
Standard Integral Formulas
These formulas appear repeatedly in exams and form the foundation for complex problems.
| Function | Integral | Important Notes |
|---|---|---|
| ∫0 dx | C | Integration of zero is constant |
| ∫1 dx | x + C | Simplest integration |
| ∫xⁿ dx | xⁿ⁺¹/(n+1) + C | n ≠ -1 |
| ∫sin x dx | -cos x + C | Negative sign is crucial |
| ∫cos x dx | sin x + C | Positive result |
| ∫sec²x dx | tan x + C | Memorize this pattern |
| ∫cosec²x dx | -cot x + C | Negative sign matters |
| ∫sec x tan x dx | sec x + C | Product form |
| ∫cosec x cot x dx | -cosec x + C | Watch the negative |
| ∫1/√(1-x²) dx | sin⁻¹x + C or -cos⁻¹x + C | Inverse trig form |
| ∫1/(1+x²) dx | tan⁻¹x + C | Very common in exams |
| ∫1/(x√(x²-1)) dx | sec⁻¹x + C | Less common but important |
| ∫eˣ dx | eˣ + C | Exponential property |
| ∫aˣ dx | aˣ/ln a + C | General exponential |
| ∫1/x dx | ln|x| + C | Natural logarithm |
Integration by Parts Formula
This is one of the most powerful integration techniques, used when the product of two functions needs integration.
Formula:
∫u·v dx = u∫v dx - ∫[du/dx · ∫v dx] dx
Or written as:
∫u dv = uv - ∫v du
When to Use Integration by Parts
Use this method when:
- Integrating a product of two different types of functions
- One function becomes simpler when differentiated
- Direct integration is not possible
ILATE Rule (Choosing u and v)
ILATE helps you decide which function to choose as 'u':
| Priority | Type | Example |
|---|---|---|
| I | Inverse trigonometric | sin⁻¹x, tan⁻¹x, cos⁻¹x |
| L | Logarithmic | log x, ln x |
| A | Algebraic | x, x², x³, polynomial |
| T | Trigonometric | sin x, cos x, tan x |
| E | Exponential | eˣ, aˣ |
How to use: The function appearing first in ILATE should be chosen as 'u'.
Example: For ∫x·sin x dx, choose u = x (algebraic) and dv = sin x dx (trigonometric).
Integration of UV Formula
The UV formula is another name for integration by parts, specifically highlighting the product nature.
Standard Form:
∫u(x)·v(x) dx = u(x)∫v(x) dx - ∫[u'(x)·∫v(x) dx] dx
Step-by-step process:
- Identify u and v using ILATE rule
- Differentiate u to get u'
- Integrate v to get ∫v dx
- Apply the formula
- Simplify the remaining integral
Common UV Integration Patterns:
| Type | Example | First Choice (u) |
|---|---|---|
| Polynomial × Exponential | x²·eˣ | x² (algebraic) |
| Polynomial × Trigonometric | x·sin x | x (algebraic) |
| Logarithmic × Polynomial | x·ln x | ln x (logarithmic) |
| Inverse Trig × Polynomial | x·sin⁻¹x | sin⁻¹x (inverse trig) |
| Exponential × Trigonometric | eˣ·sin x | Either works (apply twice) |
Trigonometric Integration Formulas
Trigonometric integrals are heavily tested in all mathematics exams.
| Function | Integral | Point |
|---|---|---|
| ∫sin x dx | -cos x + C | Negative sign |
| ∫cos x dx | sin x + C | Positive sign |
| ∫tan x dx | ln|sec x| + C or -ln|cos x| + C | Both forms correct |
| ∫cot x dx | ln|sin x| + C | Logarithmic result |
| ∫sec x dx | ln|sec x + tan x| + C | Difficult to derive |
| ∫cosec x dx | ln|cosec x - cot x| + C | Often forgotten |
| ∫sec²x dx | tan x + C | Direct result |
| ∫cosec²x dx | -cot x + C | Watch negative |
| ∫sec x tan x dx | sec x + C | Product form |
| ∫cosec x cot x dx | -cosec x + C | Product form |
| ∫sin²x dx | (x/2) - (sin 2x)/4 + C | Use double angle formula |
| ∫cos²x dx | (x/2) + (sin 2x)/4 + C | Use double angle formula |
Pro tip: For sin²x and cos²x, remember the identities:
- sin²x = (1 - cos 2x)/2
- cos²x = (1 + cos 2x)/2
Inverse Trigonometric Integration Formulas
These formulas are essential for competitive exams and appear frequently.
| Function | Integral | Condition |
|---|---|---|
| ∫1/√(a²-x²) dx | sin⁻¹(x/a) + C | |x| < a |
| ∫1/√(x²-a²) dx | ln|x + √(x²-a²)| + C | |x| > a |
| ∫1/(a²+x²) dx | (1/a)tan⁻¹(x/a) + C | All x |
| ∫1/(x²-a²) dx | (1/2a)ln|(x-a)/(x+a)| + C | x ≠ ±a |
| ∫1/(x√(x²-a²)) dx | (1/a)sec⁻¹(x/a) + C | |x| > a |
| ∫1/(x√(a²-x²)) dx | -(1/a)ln|(a+√(a²-x²))/x| + C | 0 < |x| < a |
Important: These formulas involve the parameter 'a'. When a = 1, formulas simplify significantly.
Definite Integral Formulas
Definite integrals calculate the exact area under a curve between two limits.
Basic Definition:
∫ₐᵇ f(x) dx = F(b) - F(a)
Where F(x) is the antiderivative of f(x).
Key Properties of Definite Integrals
| Property | Formula | Use Case |
|---|---|---|
| Reversing limits | ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx | Changing integration direction |
| Zero width | ∫ₐᵃ f(x) dx = 0 | Same upper and lower limit |
| Splitting interval | ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx | Breaking complex integrals |
| Constant multiple | ∫ₐᵇ k·f(x) dx = k·∫ₐᵇ f(x) dx | Taking constants out |
| Sum rule | ∫ₐᵇ [f(x)+g(x)] dx = ∫ₐᵇ f(x) dx + ∫ₐᵇ g(x) dx | Separating functions |
| Symmetry (even) | ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx | When f(-x) = f(x) |
| Symmetry (odd) | ∫₋ₐᵃ f(x) dx = 0 | When f(-x) = -f(x) |
King Property (Most Important for Exams)
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx
Application: This property simplifies many complex definite integrals dramatically.
Reduction Formula Integration
Reduction formulas help solve integrals of the form ∫xⁿ·f(x) dx by reducing the power n step by step.
General Concept: Express Iₙ (integral with power n) in terms of Iₙ₋₁ or Iₙ₋₂.
Common Reduction Formulas
1. For ∫sinⁿx dx:
Iₙ = ∫sinⁿx dx = -(sinⁿ⁻¹x·cos x)/n + [(n-1)/n]Iₙ₋₂
2. For ∫cosⁿx dx:
Iₙ = ∫cosⁿx dx = (cosⁿ⁻¹x·sin x)/n + [(n-1)/n]Iₙ₋₂
3. For ∫tanⁿx dx:
Iₙ = ∫tanⁿx dx = (tanⁿ⁻¹x)/(n-1) - Iₙ₋₂
4. For ∫secⁿx dx:
Iₙ = ∫secⁿx dx = (secⁿ⁻²x·tan x)/(n-1) + [(n-2)/(n-1)]Iₙ₋₂
5. For ∫xⁿ·eˣ dx:
Iₙ = xⁿ·eˣ - n·Iₙ₋₁
When to use: When dealing with high powers of trigonometric or exponential functions multiplied by polynomials.
Differentiation and Integration Formula Relationship
Integration and differentiation are inverse operations. Understanding this relationship helps verify answers.
| Differentiation | Integration |
|---|---|
| d/dx(xⁿ) = n·xⁿ⁻¹ | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C |
| d/dx(sin x) = cos x | ∫cos x dx = sin x + C |
| d/dx(cos x) = -sin x | ∫sin x dx = -cos x + C |
| d/dx(tan x) = sec²x | ∫sec²x dx = tan x + C |
| d/dx(eˣ) = eˣ | ∫eˣ dx = eˣ + C |
| d/dx(ln x) = 1/x | ∫(1/x) dx = ln|x| + C |
| d/dx(sin⁻¹x) = 1/√(1-x²) | ∫1/√(1-x²) dx = sin⁻¹x + C |
| d/dx(tan⁻¹x) = 1/(1+x²) | ∫1/(1+x²) dx = tan⁻¹x + C |
Verification Method:
After integrating, differentiate your answer. You should get back the original function.
Advanced Integration Formulas
These formulas appear in competitive exams and advanced problems.
Cauchy Integral Formula (Complex Analysis)
For a function f(z) analytic inside and on a simple closed contour C:
f(a) = (1/2πi) ∮_C [f(z)/(z-a)] dz
Derivative form:
f⁽ⁿ⁾(a) = (n!/2πi) ∮_C [f(z)/(z-a)ⁿ⁺¹] dz
Note: This is typically covered in higher mathematics courses but appears in some advanced JEE problems.
Leibniz Integral Rule
For differentiating under the integral sign:
d/dt[∫ₐ₍ₜ₎^b₍ₜ₎ f(x,t) dx] = ∫ₐ₍ₜ₎^b₍ₜ₎ (∂f/∂t) dx + f(b,t)·b'(t) - f(a,t)·a'(t)
Integration by Substitution
When integrating f(g(x))·g'(x):
∫f(g(x))·g'(x) dx = ∫f(u) du [where u = g(x)]
Common substitutions:
| Expression Type | Substitution | Why It Works |
|---|---|---|
| √(a²-x²) | x = a sin θ | Converts to trigonometric |
| √(a²+x²) | x = a tan θ | Simplifies radical |
| √(x²-a²) | x = a sec θ | Removes square root |
| 1/(ax+b) | t = ax+b | Linear substitution |
Solved Examples
Example 1: Basic Power Rule
Problem: ∫x⁵ dx
Solution: Using ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Here n = 5
∫x⁵ dx = x⁶/6 + C
Answer: x⁶/6 + C
Example 2: Integration by Parts
Problem: ∫x·eˣ dx
Solution: Step 1: Choose u and v using ILATE
- u = x (Algebraic)
- dv = eˣ dx (Exponential)
Step 2: Find du and v
- du = dx
- v = eˣ
Step 3: Apply formula: ∫u dv = uv - ∫v du
∫x·eˣ dx = x·eˣ - ∫eˣ dx = x·eˣ - eˣ + C = eˣ(x - 1) + C
Answer: eˣ(x - 1) + C
Example 3: Trigonometric Integration
Problem: ∫sin²x dx
Solution: Step 1: Use identity sin²x = (1 - cos 2x)/2
∫sin²x dx = ∫(1 - cos 2x)/2 dx = (1/2)∫(1 - cos 2x) dx = (1/2)[x - (sin 2x)/2] + C = x/2 - (sin 2x)/4 + C
Answer: x/2 - (sin 2x)/4 + C
Example 4: Definite Integral Using Symmetry
Problem: ∫₋₂² x³ dx
Solution: Notice that f(x) = x³ is an odd function because f(-x) = -x³ = -f(x)
For odd functions: ∫₋ₐᵃ f(x) dx = 0
Answer: 0
Example 5: Substitution Method
Problem: ∫2x·e^(x²) dx
Solution: Step 1: Let u = x² Step 2: Then du = 2x dx Step 3: The integral becomes ∫eᵘ du = eᵘ + C Step 4: Substitute back: e^(x²) + C
Answer: e^(x²) + C
Common Mistakes to Avoid
Mistake 1: Forgetting the Constant of Integration
Wrong: ∫x dx = x²/2
Right: ∫x dx = x²/2 + C
Why it matters: Without C, your answer is incomplete and will lose marks.
Mistake 2: Sign Errors in Trigonometric Integration
Wrong: ∫sin x dx = cos x + C
Right: ∫sin x dx = -cos x + C
Memory tip: Sine integral goes negative, cosine stays positive.
Mistake 3: Wrong Power Rule Application
Wrong: ∫(1/x) dx = x⁰/0 + C (undefined!)
Right: ∫(1/x) dx = ln|x| + C
Remember: Power rule doesn't work when n = -1.
Mistake 4: Incorrect ILATE Application
Wrong: For ∫x·ln x dx, choosing u = x
Right: Choose u = ln x (logarithmic comes before algebraic)
Result: Correct choice makes the problem much simpler.
Mistake 5: Forgetting Absolute Value in Logarithms
Wrong: ∫(1/x) dx = ln x + C
Right: ∫(1/x) dx = ln|x| + C
Why: ln is undefined for negative x, but the integral exists for all x ≠ 0.
Mistake 6: Ignoring Limits in Definite Integrals
When finding ∫₁³ x² dx:
Wrong: = x³/3 (just the antiderivative)
Right: = [x³/3]₁³ = 27/3 - 1/3 = 26/3
Remember: Always substitute both limits and subtract.
Tips and Tricks
Trick 1: LIATE vs ILATE
Both acronyms work. Use whichever you remember better:
- ILATE: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential
- LIATE: Logarithmic, Inverse, Algebraic, Trigonometric, Exponential
Trick 2: Quick Check Using Differentiation
After integrating, differentiate your answer. You should get back the original function.
Example: If ∫2x dx = x² + C, then d/dx(x² + C) = 2x
Trick 3: Memorize Pairs
Learn differentiation and integration as pairs:
- sin x ↔ cos x
- eˣ ↔ eˣ
- xⁿ ↔ xⁿ⁺¹/(n+1)
Trick 4: Standard Form Recognition
Practice recognizing standard forms quickly:
- See 1/√(1-x²)? Think sin⁻¹x
- See 1/(1+x²)? Think tan⁻¹x
- See sec²x? Think tan x
Trick 5: For Definite Integrals, Check for Symmetry First
Before calculating, check:
- Even function? Use ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx
- Odd function? Answer is 0!
This saves enormous calculation time in exams.
Trick 6: Create a Formula Sheet
Make a one-page formula sheet with:
- All basic integrals
- Integration by parts formula
- Standard substitutions
- Trigonometric identities
Revise this sheet daily one week before exams.
FAQs about Integration Formulas
Q. What is the integration by parts formula?
The integration by parts formula is ∫u dv = uv - ∫v du. It's used for integrating products of two functions. Choose u using the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) and dv as the remaining part.
Q. How many integration formulas are there in Class 12?
Class 12 covers approximately 25-30 basic integration formulas including power rule, trigonometric integrals, exponential functions, and inverse trigonometric forms. However, with variations and special cases, you should know around 50 formulas for comprehensive exam preparation.
Q. What is the difference between definite and indefinite integrals?
Indefinite integrals have no limits and include a constant C (∫f(x)dx = F(x) + C). Definite integrals have upper and lower limits and give a specific numerical value (∫ₐᵇf(x)dx = F(b) - F(a)). Definite integrals represent actual areas.
Q. What is the ILATE rule in integration?
ILATE is a mnemonic for choosing which function to differentiate (u) in integration by parts: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. The function appearing first in ILATE should be chosen as u, and the other as dv.
Q. Why do we add constant C in integration?
We add C because differentiation of any constant is zero. When finding an antiderivative, infinite functions differ by a constant. For example, both x² and x²+5 have derivative 2x, so the general antiderivative is x²+C.
Q. How to remember integration formulas easily?
Practice daily, create a formula sheet, learn formulas in pairs with differentiation, use mnemonics like ILATE, understand the logic behind formulas rather than rote learning, solve 5-10 problems daily, and revise formulas before sleeping for better retention.
Q. What is reduction formula in integration?
Reduction formulas express an integral with power n (Iₙ) in terms of a similar integral with lower power (Iₙ₋₁ or Iₙ₋₂). They're used for integrals like ∫sinⁿx dx or ∫xⁿeˣ dx where direct integration is difficult.
Q. Can I use integration formulas in JEE and NEET?
Yes, integration formulas are essential for JEE Main, JEE Advanced, and NEET. JEE Advanced requires mastery of reduction formulas and advanced techniques. NEET focuses more on basic integration and applications to physics problems. All standard formulas are allowed and expected.
Conclusion
Integration is the backbone of calculus and forms a critical component of Class 12 Mathematics and competitive exams. This comprehensive guide has covered over 50 essential formulas—from basic power rules to advanced reduction formulas.
- Master the 15-20 basic formulas first
- Use ILATE for integration by parts
- Practice symmetry properties for definite integrals
- Verify answers by differentiating
- Create and revise a personal formula sheet
Remember, integration is not about memorizing formulas blindly. Understanding when and how to apply each formula makes the difference between average and excellent performance.
The formulas in this guide represent years of mathematical development. They're tools that will serve you not just in exams but in higher studies and real-world applications in engineering, physics, economics, and data science.