Integral Formulas are the foundational rules used to evaluate indefinite and definite integrals in mathematics, forming a core part of CBSE Class 12 Maths (Chapter 7 — Integrals) and carrying significant weightage in JEE Main, JEE Advanced, and NEET 2025-26 examinations. These formulas allow students to find antiderivatives of algebraic, trigonometric, exponential, and logarithmic functions quickly and accurately. This article covers every essential integral formula, a complete formula sheet, three progressive solved examples, CBSE exam tips, common mistakes, and JEE/NEET application strategies.
Key Integral Formulas at a Glance
Quick reference for the most important integral formulas used in CBSE Class 12 and competitive exams.
- Power Rule: \( \int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C, \quad n \neq -1 \)
- Reciprocal: \( \int \dfrac{1}{x} \, dx = \ln|x| + C \)
- Exponential: \( \int e^x \, dx = e^x + C \)
- Sine: \( \int \sin x \, dx = -\cos x + C \)
- Cosine: \( \int \cos x \, dx = \sin x + C \)
- Integration by Parts: \( \int u \, dv = uv – \int v \, du \)
- Definite Integral: \( \int_a^b f(x)\, dx = F(b) – F(a) \)
What are Integral Formulas?
Integral Formulas are pre-derived mathematical rules that give the antiderivative (or integral) of standard functions directly. Integration is the reverse process of differentiation. When we differentiate \( F(x) \), we get \( f(x) \). Conversely, integrating \( f(x) \) returns \( F(x) + C \), where \( C \) is the constant of integration.
In NCERT Class 12 Mathematics, Chapter 7 (Integrals) introduces two types of integrals. The first is the indefinite integral, written as \( \int f(x)\, dx \). The second is the definite integral, written as \( \int_a^b f(x)\, dx \). Both types rely on a standard set of integral formulas that students must memorise for CBSE board exams and competitive tests.
These formulas cover power functions, trigonometric functions, inverse trigonometric functions, exponential functions, and logarithmic functions. Mastering integral formulas enables students to solve area-under-curve problems, differential equations, and physics-based calculus problems with confidence. The NCERT textbook lists these formulas in the formulae section of Chapter 7, and they are directly applicable in CBSE board questions as well as JEE and NEET papers.
Integral Formulas — Expression and Variables
The general form of an indefinite integral is expressed as:
\[ \int f(x)\, dx = F(x) + C \]
Here, \( f(x) \) is called the integrand, \( F(x) \) is the antiderivative, \( dx \) indicates integration with respect to \( x \), and \( C \) is the arbitrary constant of integration.
The general form of a definite integral is:
\[ \int_a^b f(x)\, dx = F(b) – F(a) \]
| Symbol | Meaning | Notes |
|---|---|---|
| \( \int \) | Integral sign | Derived from the letter “S” (summa) |
| \( f(x) \) | Integrand | The function being integrated |
| \( dx \) | Differential of x | Indicates the variable of integration |
| \( F(x) \) | Antiderivative | \( F'(x) = f(x) \) |
| \( C \) | Constant of integration | Arbitrary constant; always added in indefinite integrals |
| \( a, b \) | Limits of integration | Lower limit \( a \), upper limit \( b \) in definite integrals |
| \( n \) | Power/exponent | Must not equal \(-1\) in the power rule |
Derivation of the Power Rule for Integration
The power rule for integration is derived directly from the power rule for differentiation. We know that:
\[ \frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right) = x^n \]
Since integration reverses differentiation, we write:
\[ \int x^n\, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1 \]
When \( n = -1 \), the formula above becomes undefined. In that case, we use the special result \( \int \frac{1}{x}\, dx = \ln|x| + C \), which is derived from the derivative of \( \ln|x| \). This exception is frequently tested in CBSE and JEE examinations.
Complete Integral Formula Sheet
The table below lists all standard integral formulas covered in NCERT Class 12 Maths and required for JEE/NEET 2025-26. Bookmark this for quick revision.
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Power Rule | \( \int x^n\, dx = \dfrac{x^{n+1}}{n+1} + C \) | n = any real number, \( n \neq -1 \) | Dimensionless (pure maths) | Class 12, Ch 7 |
| Reciprocal Rule | \( \int \dfrac{1}{x}\, dx = \ln|x| + C \) | x ≠ 0 | Dimensionless | Class 12, Ch 7 |
| Exponential (base e) | \( \int e^x\, dx = e^x + C \) | x = variable | Dimensionless | Class 12, Ch 7 |
| Exponential (base a) | \( \int a^x\, dx = \dfrac{a^x}{\ln a} + C \) | a > 0, a ≠ 1 | Dimensionless | Class 12, Ch 7 |
| Sine | \( \int \sin x\, dx = -\cos x + C \) | x in radians | Dimensionless | Class 12, Ch 7 |
| Cosine | \( \int \cos x\, dx = \sin x + C \) | x in radians | Dimensionless | Class 12, Ch 7 |
| Secant Squared | \( \int \sec^2 x\, dx = \tan x + C \) | x in radians | Dimensionless | Class 12, Ch 7 |
| Cosecant Squared | \( \int \csc^2 x\, dx = -\cot x + C \) | x in radians | Dimensionless | Class 12, Ch 7 |
| Sec·Tan | \( \int \sec x \tan x\, dx = \sec x + C \) | x in radians | Dimensionless | Class 12, Ch 7 |
| Csc·Cot | \( \int \csc x \cot x\, dx = -\csc x + C \) | x in radians | Dimensionless | Class 12, Ch 7 |
| Inverse Sine | \( \int \dfrac{1}{\sqrt{1-x^2}}\, dx = \sin^{-1} x + C \) | |x| < 1 | Dimensionless | Class 12, Ch 7 |
| Inverse Tangent | \( \int \dfrac{1}{1+x^2}\, dx = \tan^{-1} x + C \) | x ∈ ℝ | Dimensionless | Class 12, Ch 7 |
| Integration by Parts | \( \int u\, dv = uv – \int v\, du \) | u, v are functions of x | Dimensionless | Class 12, Ch 7 |
| Definite Integral (Newton–Leibniz) | \( \int_a^b f(x)\, dx = F(b) – F(a) \) | a = lower limit, b = upper limit | Depends on context | Class 12, Ch 7 |
| Constant Multiple Rule | \( \int k \cdot f(x)\, dx = k \int f(x)\, dx \) | k = constant | Dimensionless | Class 12, Ch 7 |
| Sum/Difference Rule | \( \int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx \) | f, g are integrable functions | Dimensionless | Class 12, Ch 7 |
Integral Formulas — Solved Examples
Example 1 (Class 9-10 Level — Basic Power Rule)
Problem: Evaluate \( \int (3x^2 + 5x – 4)\, dx \).
Given: Integrand \( f(x) = 3x^2 + 5x – 4 \)
Step 1: Apply the Sum/Difference Rule to split the integral:
\( \int (3x^2 + 5x – 4)\, dx = \int 3x^2\, dx + \int 5x\, dx – \int 4\, dx \)
Step 2: Apply the Constant Multiple Rule and Power Rule to each term:
\( = 3 \cdot \dfrac{x^3}{3} + 5 \cdot \dfrac{x^2}{2} – 4x + C \)
Step 3: Simplify each term:
\( = x^3 + \dfrac{5x^2}{2} – 4x + C \)
Answer
\( \int (3x^2 + 5x – 4)\, dx = x^3 + \dfrac{5x^2}{2} – 4x + C \)
Example 2 (Class 11-12 Level — Integration by Parts)
Problem: Evaluate \( \int x e^x\, dx \).
Given: Integrand \( f(x) = x e^x \). Use Integration by Parts: \( \int u\, dv = uv – \int v\, du \).
Step 1: Choose \( u \) and \( dv \) using the ILATE rule (Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential). Here, \( u = x \) (Algebraic) and \( dv = e^x\, dx \) (Exponential).
Step 2: Find \( du \) and \( v \):
\( du = dx \) and \( v = e^x \)
Step 3: Apply the Integration by Parts formula:
\( \int x e^x\, dx = x \cdot e^x – \int e^x\, dx \)
Step 4: Evaluate the remaining integral:
\( = x e^x – e^x + C \)
Step 5: Factor for a cleaner form:
\( = e^x(x – 1) + C \)
Answer
\( \int x e^x\, dx = e^x(x – 1) + C \)
Example 3 (JEE/NEET Level — Definite Integral with Substitution)
Problem: Evaluate \( \displaystyle\int_0^{\pi/2} \sin^2 x\, dx \).
Given: Definite integral with limits \( a = 0 \) and \( b = \pi/2 \).
Step 1: Use the trigonometric identity to reduce \( \sin^2 x \):
\( \sin^2 x = \dfrac{1 – \cos 2x}{2} \)
Step 2: Rewrite the integral:
\( \displaystyle\int_0^{\pi/2} \sin^2 x\, dx = \int_0^{\pi/2} \dfrac{1 – \cos 2x}{2}\, dx \)
Step 3: Split and integrate term by term:
\( = \dfrac{1}{2}\int_0^{\pi/2} 1\, dx – \dfrac{1}{2}\int_0^{\pi/2} \cos 2x\, dx \)
Step 4: Evaluate each integral:
\( = \dfrac{1}{2}\left[x\right]_0^{\pi/2} – \dfrac{1}{2}\left[\dfrac{\sin 2x}{2}\right]_0^{\pi/2} \)
Step 5: Substitute the limits:
\( = \dfrac{1}{2} \cdot \dfrac{\pi}{2} – \dfrac{1}{4}\left[\sin \pi – \sin 0\right] \)
\( = \dfrac{\pi}{4} – \dfrac{1}{4}(0 – 0) = \dfrac{\pi}{4} \)
Answer
\( \displaystyle\int_0^{\pi/2} \sin^2 x\, dx = \dfrac{\pi}{4} \)
CBSE Exam Tips 2025-26
- Memorise the formula sheet: CBSE Class 12 board papers (2025-26) directly test standard integral formulas. Write them out daily for two weeks before the exam.
- Always add the constant C: In indefinite integrals, omitting \( + C \) costs half a mark in CBSE marking schemes. We recommend writing \( + C \) as a reflex habit.
- Use the ILATE rule for Integration by Parts: This rule (Inverse trig → Log → Algebraic → Trig → Exponential) helps you choose \( u \) correctly every time. It prevents the most common IBP error.
- Verify using differentiation: After evaluating an indefinite integral, differentiate your answer. If you get back the original integrand, your answer is correct. This is a powerful self-check strategy.
- Know the definite integral properties: Properties such as \( \int_a^b f(x)\, dx = \int_a^b f(a+b-x)\, dx \) and \( \int_0^{2a} f(x)\, dx \) shortcuts appear in almost every CBSE board paper. Our experts suggest practising at least 10 property-based problems before the exam.
- Time management: Integration questions carry 4-6 marks each in CBSE 2025-26. Allocate 6-8 minutes per question. If a substitution does not simplify within two steps, try a different method.
Common Mistakes to Avoid
- Forgetting the constant of integration: Every indefinite integral must end with \( + C \). Students lose marks in CBSE and JEE for omitting it. The constant represents an entire family of antiderivatives.
- Applying the power rule when n = −1: The formula \( \int x^n\, dx = \frac{x^{n+1}}{n+1} + C \) fails when \( n = -1 \). The correct result is \( \int \frac{1}{x}\, dx = \ln|x| + C \). Using the power rule here gives a division-by-zero error.
- Incorrect sign in trigonometric integrals: A very common error is writing \( \int \sin x\, dx = \cos x + C \) instead of \( -\cos x + C \). Always recall that the derivative of \( \cos x \) is \( -\sin x \), confirming the negative sign.
- Wrong choice of u and dv in Integration by Parts: Choosing the exponential function as \( u \) instead of the algebraic term leads to a more complex integral. Always follow the ILATE priority order.
- Not changing limits in definite integrals after substitution: When using substitution in a definite integral, the limits must also be transformed in terms of the new variable. Failing to do so is one of the most penalised errors in JEE Main.
JEE/NEET Application of Integral Formulas
Integral Formulas appear extensively in both JEE Main and JEE Advanced papers. In our experience, JEE aspirants who master the standard formula list and their combinations consistently score higher in the Calculus section, which carries approximately 20-25% of the mathematics paper.
Pattern 1 — Direct Formula Application (JEE Main)
JEE Main frequently tests direct substitution into standard integral formulas. A typical question asks students to evaluate an integral of the form \( \int \frac{dx}{a^2 + x^2} \). The answer is \( \frac{1}{a}\tan^{-1}\frac{x}{a} + C \). Recognising the standard form instantly saves 2-3 minutes per question.
Pattern 2 — Definite Integral Properties (JEE Advanced)
JEE Advanced tests properties of definite integrals deeply. The King’s Property, \( \int_a^b f(x)\, dx = \int_a^b f(a+b-x)\, dx \), is used in nearly every JEE Advanced paper. It simplifies complex integrals involving \( \sin x + \cos x \) combinations to straightforward calculations. In our experience, this single property solves 3-4 questions per paper.
Pattern 3 — Area Under Curves (NEET)
NEET Physics and Biology do not test integration directly. However, NEET Physics uses the concept of area under a velocity-time graph (which is displacement) and area under a force-displacement graph (which is work done). These are definite integral applications. Understanding \( \int_a^b f(x)\, dx = F(b) – F(a) \) as an area concept is therefore critical for NEET Physics numericals involving kinematics and work-energy theorem.
We recommend that JEE aspirants solve at least 50 integration problems per chapter-revision cycle. Focus specifically on integration by partial fractions and integration by substitution, as these are the two most commonly tested advanced techniques in JEE Main 2025.
FAQs on Integral Formulas
Explore More Formula Resources
Strengthen your mathematics preparation with these related formula guides on ncertbooks.net. For a complete overview of all algebraic and mathematical formulas, visit our Algebra Formulas hub.
If you are studying trigonometry alongside integration, our Cotangent Formula guide covers all trigonometric ratios and their integral forms in detail. Students preparing for CBSE Class 10 and Class 11 will also benefit from our Axis of Symmetry Formula article, which connects coordinate geometry with calculus concepts. For compound growth problems that use integral-like reasoning, explore our Daily Compound Interest Formula guide.
For official NCERT textbook content on Integrals, refer to the NCERT Class 12 Maths Chapter 7 on ncert.nic.in.