Stepwise Methods to Solve Definite & Indefinite Integration Problems
The concept of Integration Questions with Solutions PDF Class 12 Maths plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It is one of the most important chapters for Class 12 students preparing for CBSE Boards, JEE Main, and other competitive exams. This page explains what integration is, key formulas, and common question types, and provides step-by-step integration solutions along with a handy PDF for offline practice and revision.
What Is Integration Questions with Solutions PDF Class 12 Maths?
Integration is the process of finding the area under a curve, or the antiderivative of a function. In Class 12 Maths, integration questions typically involve calculating definite integrals (with limits), indefinite integrals (without limits), and applying methods like substitution, integration by parts, and using partial fractions. You’ll find this concept applied in geometry (finding areas and volumes), physics (calculating displacement, work, or probability), and engineering problems.
Key Formula for Integration Questions with Solutions PDF Class 12 Maths
Here are some essential integration formulas for Class 12:
| Integral | Formula | Notes |
|---|---|---|
| \(\int x^n dx\) | \(\frac{x^{n+1}}{n+1} + C\) (n ≠ -1) | Basic Power Rule |
| \(\int e^x dx\) | \(e^x + C\) | Exponential Function |
| \(\int \sin x dx\) | \(-\cos x + C\) | Trigonometric |
| \(\int \cos x dx\) | \(\sin x + C\) | Trigonometric |
| \(\int \frac{1}{x} dx\) | \(\ln|x| + C\) | Logarithmic Function |
Cross-Disciplinary Usage
Integration is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in questions on displacement, area under graphs, and even statistics.
Types of Integration Questions in Class 12 Maths
Integration questions for Class 12 Maths (CBSE, NCERT, state boards, or JEE) generally fall into these categories:
- Indefinite integrals (basic, standard forms, or algebraic manipulation)
- Definite integrals (with upper and lower limits)
- Integration by substitution (changing the variable to simplify)
- Integration by parts (product of two functions)
- Partial fractions (breaking rational expressions)
Download: Integration Questions with Solutions PDF
To boost your offline revision and exam practice, you can quickly download Integration Questions with Solutions PDF for Class 12 Maths from Vedantu. This PDF includes question sets based on all major methods, with stepwise answers and explanations tailored for CBSE, JEE Main, and competitive exams. Print or save to your phone for last-minute preparation!
Step-by-Step Illustration
Here’s how to solve one of the most common integration questions seen in Class 12 exams:
Example: Find \(\int 2x^3 dx\)
1. Start with the given integral: \(\int 2x^3 dx\)2. Bring the constant out: \(2 \int x^3 dx\)
3. Use the power rule: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\)
4. Apply it: \(2 \cdot \frac{x^{3+1}}{3+1} + C = 2 \cdot \frac{x^4}{4} + C = \frac{x^4}{2} + C\)
5. Final Answer: \(\int 2x^3 dx = \frac{x^4}{2} + C\)
Let’s try a substitution example commonly given in JEE:
Example: Find \(\int 2x \cos(x^2) dx\)
1. Let \(u = x^2\) → \(du = 2x dx\)2. Substitute: \(\int \cos(u) du\)
3. Integrate: \(\sin u + C\)
4. Substitute back: \(\sin(x^2) + C\)
5. Final Answer: \(\int 2x \cos(x^2) dx = \sin(x^2) + C\)
Speed Trick or Vedic Shortcut
For some forms, like integrating rational functions with quadratic denominators, learn the “complete the square” method for faster substitution. For definite integrals with symmetric limits, always check if the function is odd or even—many odd functions directly integrate to zero across symmetric intervals.
Example Trick: If \( f(x) \) is odd, \(\int\limits_{-a}^{a} f(x) dx = 0\). Use this trick in MCQs for faster calculation!
Try These Yourself
- Integrate \( \int (3x^2 + 4) dx \)
- Evaluate \( \int\limits_{0}^{\pi} \sin x dx \)
- Integrate by parts: \( \int x e^x dx \)
- Break into partial fractions and integrate: \( \int \frac{1}{x^2-1} dx \)
- Try a substitution for: \( \int \frac{2x}{1+x^2} dx \)
Frequent Errors and Misunderstandings
- Forgetting the constant of integration (C) in indefinite integrals
- Not converting limits when changing the variable in definite integrals
- Confusing substitution and by parts steps
- Incorrect algebraic factorization in partial fractions
Relation to Other Concepts
The idea of integration connects with differentiation (it is the reverse operation), area calculations, and advanced calculus. Understanding integration also helps in mastering applications of integrals in physics and engineering.
Classroom Tip
A quick way to remember integration formulas is to use a formula sheet before practice. Vedantu’s teachers recommend solving at least 10 mixed integration questions daily for a month before exams. Watch methods of integration video explainers to boost confidence with step-based doubt resolution.
We explored Integration Questions with Solutions PDF Class 12 Maths — from definition, types, key formulas, tricks, and solved examples. Continue revising and practicing via Vedantu’s stepwise solutions for top marks in board and entrance exams!
Related Vedantu Resources
- Definite Integral (Solved Examples)
- Application of Integrals
- Methods of Integration (Tricks Guide)
- Integration by Parts Rule
FAQs on Class 12 Integration Questions with Solutions (PDF)
1. What are the most frequently asked types of integration questions in CBSE Class 12 board exams?
For the CBSE Class 12 board exams, questions on integration are typically focused on a few key areas. Expect questions on:
- Integration by Substitution: Usually for 2-3 marks, requiring you to identify the correct substitution.
- Integration by Parts: Often appears as a 3 or 5-mark question, testing your application of the ILATE rule.
- Integration by Partial Fractions: Common in the long answer section (3-5 marks), testing your algebraic skills.
- Properties of Definite Integrals: These are very important and frequently form the basis of 5-mark questions, especially the 'King's Property' and the even/odd function property.
2. Which integration formulas carry the most weightage for solving board exam problems?
To score well, you must master the fundamental formulas. The most critical ones for the 2025-26 exams include the standard integrals of xⁿ, eˣ, aˣ, 1/x, sin(x), cos(x), sec²(x), and the special integrals for functions like 1/(x²+a²), 1/√(x²-a²), and 1/√(a²-x²). Additionally, the Integration by Parts (UV rule) formula, ∫u·v dx = u∫v dx - ∫(u'∫v dx) dx, is essential for solving long-answer questions.
3. What are the most common mistakes students make in integration that lead to losing marks in exams?
Students often lose marks in integration due to a few common errors. Be careful to avoid these:
- Forgetting the constant of integration ('+C') in indefinite integrals. This is a conceptual error and almost always results in a marks deduction.
- Failing to change the limits of integration when using the substitution method in definite integrals.
- Incorrectly choosing 'u' and 'v' when applying integration by parts (not following the ILATE rule).
- Making algebraic mistakes during the decomposition process in partial fractions.
4. What is the best strategy to identify the correct substitution for a complex integral during an exam?
A reliable strategy is to look for a function-and-its-derivative pair within the integral. Scan the integrand for a composite function, say f(g(x)). If the derivative of the inner function, g'(x), is also present as a factor in the integrand, then substituting u = g(x) is usually the correct approach. Practising this pattern recognition is key to improving speed and accuracy in exams.
5. Which properties of definite integrals are essential for solving important questions in the CBSE 2025-26 exam?
For the Class 12 board exam, a few properties of definite integrals are repeatedly tested and are crucial for solving complex problems efficiently. The most important ones to master are:
- P₀: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt
- P₂: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫cᵇ f(x) dx
- P₄ (King's Property): ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx
- P₇ (Odd/Even Functions): ∫₋ₐᵃ f(x) dx, which equals 2∫₀ᵃ f(x) dx if f(x) is even, and 0 if f(x) is odd.
6. Why is forgetting the constant of integration '+C' considered a critical error in board exams?
Forgetting the '+C' is a critical conceptual error because the integral of a function is not a single function, but a family of functions (or curves) that have the same derivative. The constant 'C' represents this entire family. Omitting it implies you have found only one specific antiderivative, which is an incomplete answer. Examiners typically deduct 0.5 to 1 mark for this mistake as it shows a gap in understanding the fundamental concept of indefinite integration.
7. How should I decide whether to use integration by parts or substitution when a problem seems confusing?
The choice depends on the structure of the integrand. Use substitution when you see a composite function and the derivative of its inner part, like in ∫2x cos(x²) dx. Use integration by parts when the integrand is a product of two distinct types of functions, such as algebraic and trigonometric (e.g., ∫x sin(x) dx). The ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) helps you decide which function to choose as 'u' for integration by parts.
8. The CBSE syllabus mentions 'definite integrals as a limit of a sum'. How important is this topic for the board exam?
While 'definite integrals as a limit of a sum' is a fundamental concept that explains the origin of definite integration, its weightage in terms of direct, high-mark questions has decreased over the years. However, you should not ignore it completely. For the CBSE 2025-26 exam, be prepared for a possible short-answer question (2-3 marks) asking you to evaluate a simple integral using this specific method. Knowing the formula and procedure is important for a comprehensive preparation.
9. What are some Higher-Order Thinking Skills (HOTS) questions related to integration that can be expected in the exam?
HOTS questions in integration go beyond direct formula application. An important example is finding the area between two intersecting curves. This requires you to:
- Find the points of intersection by solving the equations of the curves.
- Correctly identify the upper and lower curve within the required interval.
- Set up the definite integral as the area of the upper curve minus the area of the lower curve.
10. When solving a complex integral, is it better to simplify the function first or apply integration rules directly?
It is almost always better to simplify the integrand algebraically first. Many complex-looking problems become significantly easier after simplification. For example, expanding a binomial, using trigonometric identities, or splitting a fraction can transform the problem into a sum of standard integrals. Applying integration rules like 'by parts' to a non-simplified function can lead to much longer and more error-prone calculations.
