NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Integrals: NCERT Solutions for Class 12 Maths Chapter 7 Integrals helps students to understand the core ideas of calculus in simple form. This chapter is very impotant for cbse class 12 board exam and also for competitive exams like JEE Exam and NEET Exam. The Class 12 NCERT solutions are prepared as per the latest CBSE Syllabus Class 12 Maths and contain step by step methods to solve all type of integrals sums. Students can also access the integrals class 12 ncert solutions pdf download for offline study, making it convienient to practice anytime. With clear explanations and worked-out solutions, mastering integration becomes much simpler.

The topic Integrals deals with finding the area under curves and helps to understand how differentiation and integration are connected. In these NCERT Solutions for Class 12 Maths, each question is explained clearly so that learners can improve their mathematical skills and problem solving approach. The class 12 integrals ncert also guide students to learn important formulas, properties and applications in real life.

NCERT Solutions for Class 12 Maths Chapter 7 – Integrals PDF Download

NCERT Solutions for Class 12 Maths Chapter 7 – Integrals PDF Download gives simple, stepwise solutions for indefinite and definite integrals to boost cbse board exam marks and basics for JEE/NEET too. These solutions follow CBSE Syllabus class 12 maths, include formulas list, and clear examples for substitution, partial fractions, and by-parts; honestly, few steps feel tricky but practice makes it easy. Use integrals class 12 ncert solutions pdf download for offline study, quick revision, and class 12 integrals ncert topic-wise learning.

Integrals Class 12 Questions with Solutions

Below are high‑yield “Integrals Class 12 Questions with Solutions” covering standard forms, substitution, partial fractions, trigonometric integrals, by-parts, and definite integrals aligned with CBSE/NCERT patterns, suitable for board and entrance practice, presented in simple steps for students.​

Question 1: ∫(3x² − 4x + 5) dx

Solution: Use power rule. ∫3x² dx = x³, ∫(−4x) dx = −2x², ∫5 dx = 5x + C, so answer is x³ − 2x² + 5x + C.

Question 2: ∫(1/(x+2)) dx

Solution: Standard form ∫(1/u) du = ln|u| + C, with u = x+2 → ln|x+2| + C.

Question 3: ∫(e^3x) dx

Solution: ∫e^ax dx = (1/a)e^ax + C, hence (1/3)e^3x + C.

Question 4: ∫(cos 5x) dx

Solution: ∫cos(ax) dx = (1/a)sin(ax) + C → (1/5)sin 5x + C.

Question 5: ∫(sec² x) dx

Solution: Standard integral gives tan x + C.

Question 6: ∫(1/(x²+9)) dx

Solution: ∫(1/(x²+a²)) dx = (1/a) arctan(x/a) + C → (1/3) arctan(x/3) + C.

Question 7: ∫(x/((x+1)(x+2))) dx by partial fractions

Solution: Decompose x/((x+1)(x+2)) = A/(x+1) + B/(x+2). Solve x = A(x+2) + B(x+1) → A + B = 1, 2A + B = 0 → A = −1, B = 2. Integrate: ∫(−1/(x+1) + 2/(x+2)) dx = −ln|x+1| + 2 ln|x+2| + C.

Question 8: ∫((2x+1)/(x²+x+1)) dx

Solution: Split numerator as derivative + remainder. d/dx(x²+x+1) = 2x+1, so integral is ln(x²+x+1) + C.

Question 9: ∫(x√(1+x²)) dx by substitution

Solution: Let u = 1 + x², du = 2x dx → (1/2)∫√u du = (1/2)·(2/3)u^3/2 + C = (1/3)(1+x²)^3/2 + C.

Question 10: ∫(ln x) dx by parts

Solution: Take u = ln x, dv = dx → du = dx/x, v = x. Then ∫ ln x dx = x ln x − ∫ x·(1/x) dx = x ln x − x + C.

Question 11: ∫(x e^x) dx by parts

Solution: u = x, dv = e^x dx → du = dx, v = e^x. Then ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C = e^x(x − 1) + C.

Question 12: ∫(sin³ x) dx

Solution: Write sin³ x = sin x(1 − cos² x). Let u = cos x, du = −sin x dx. Then ∫ sin x(1 − cos² x) dx = −∫(1 − u²) du = −(u − u³/3) + C = −cos x + (cos³ x)/3 + C.

Question 13: ∫(sec x tan x) dx

Solution: Derivative of sec x is sec x tan x, so answer is sec x + C.

Question 14: ∫(dx/(x√(x² − 1))) for |x|>1

Solution: Standard: ∫ dx/(x√(x² − a²)) = (1/a) arcsec(|x|/a) + C with a=1, or use substitution x = sec t to get arcsec|x| + C.

Question 15: ∫(dx/(1 + x²)) from 0 to 1

Solution: Definite integral: [arctan x]₀¹ = π/4 − 0 = π/4.

Partial Fractions and Rational Forms

Question 16: ∫((2x)/(x²+3x+2)) dx

Solution: Factor denominator (x+1)(x+2). Decompose 2x/((x+1)(x+2)) = A/(x+1)+B/(x+2). Solve 2x = A(x+2)+B(x+1) → A+B=2, 2A+B=0 → A = −2, B = 4. Integrate to get −2 ln|x+1| + 4 ln|x+2| + C.

Question 17: ∫(dx/(x² − 9))

Solution: Use partial fractions: 1/(x²−9) = (1/6)(1/(x−3) − 1/(x+3)). Integrate to (1/6)[ln|x−3| − ln|x+3|] + C = (1/6) ln|(x−3)/(x+3)| + C.

Question 18: ∫((3x − 1)/((x − 1)(x − 2)(x − 3))) dx

Solution: Decompose into A/(x−1)+B/(x−2)+C/(x−3). Solve constants and integrate termwise to logs; this is a standard Ex 7.5 type. Final form: A ln|x−1| + B ln|x−2| + C ln|x−3| + C.

By Parts and Classic Trigonometric Forms

Question 19: ∫ x sin x dx

Solution: u = x, dv = sin x dx → du = dx, v = −cos x. Then ∫ x sin x dx = −x cos x + ∫ cos x dx = −x cos x + sin x + C.

Question 20: ∫(dx/(a² + x²)) from 0 to a, a>0

Solution: Indefinite ∫ dx/(a²+x²) = (1/a) arctan(x/a) + C. Apply limits 0→a: (1/a)[arctan(1) − arctan(0)] = (1/a)(π/4 − 0) = π/(4a).

Class 12 Maths NCERT Solutions Chapter 7 Integrals

Class 12 Maths Chapter 7 on Integrals includes important concepts and exercises explained in simple steps:

• 7.1 Introduction: Basic idea of integrals and their importance.
• 7.2 Integration as Inverse of Differentiation: Shows integration as undoing differentiation.
• 7.2.1 Geometrical Interpretation: Explains area under curve concept via indefinite integrals.
• 7.2.2 Properties of Indefinite Integrals: Lists rules like linearity.
• 7.2.3 Comparison: Highlights differences and similarities of differentiation and integration.
• 7.3 Methods of Integration: Techniques to solve integrals.
• 7.3.1 Substitution: Changing variable to simplify integral.
• 7.3.2 Trigonometric Identities: Use trig formulas to integrate trig functions.
• 7.4 Integrals of Some Functions: Integration of standard functions.
• 7.5 Partial Fractions: Breaking complex fractions for easier integration.
• 7.6 Integration by Parts: Applying product rule in reverse to integrate.
• 7.7 Definite Integral: Integration with limits, calculating exact areas.
• 7.7.1 Limit of a Sum: Shows definite integral as sum of small parts.
• 7.8 Fundamental Theorem of Calculus: Connects integration and differentiation.
• 7.8.1 Area Function: Area under curve as function of limit.
• 7.8.2 First Fundamental Theorem: Differentiation of area function equals integrand.
• 7.8.3 Second Fundamental Theorem: Definite integral found from antiderivatives.
• 7.9 Evaluating Definite Integrals by Substitution: Simplifying definite integrals using substitution.
• 7.10 Properties of Definite Integrals: Important rules like additivity and reversal of limits.

NCERT Solutions for Class 12 Maths Chapter 7 - Integrals

The chapter Integrals is an important part of Calculus in the CBSE syllabus for Class 12, usually taught in term II and carrying around 35 marks in exams. This chapter has multiple activities to help students understand integrals better. The main topics covered include the inverse relation between differentiation and integration, properties of indefinite integrals, and standard integration methods like substitution, partial fractions, and integration by parts.

1. Understanding integration as the opposite of differentiation and its basic concepts.
2. The geometric meaning of indefinite integrals as a family curves shifted vertically.
3. Key properties and common formulas of indefinite integrals.
4. Methods like integration by substitution and partial fractions.
5. Integration by parts technique explained with examples.
6. The Fundamental Theorems of Integral Calculus.
7. Definite integrals explained as the limit of a sum along with their properties.

Using NCERT Solutions for Class 12 Maths Chapter 7 helps students solve difficult integrals questions, clear doubts, prepare well for exams, and revise all integral concepts with confidence. These solutions are designed as per the latest CBSE Syllabus Class 12 Maths and cover topics like indefinite and definite integrals, substitution, and more, making calculus easier to understand, though some parts might seem a bit tricky initially but consistent practice helps a lot.

Key Features of NCERT Solutions for Class 12 Maths Chapter 7 - Integrals

• Comprehensive explanations of integration methods and properties.
• Step-by-step approach for substitution, partial fractions, and by parts integration.
• Detailed coverage of Fundamental Theorems of Calculus and definite integrals.
• Practice with typical exam questions as per CBSE guidelines.