Master Integrals – Class 12 Maths Chapter 7 | Indefinite & Definite Integrals, Techniques & CBSE PYQs

 Here’s the comprehensive guide for Chapter 7: Integrals from Class 12 CBSE Maths:

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🧠 A. Key Concepts & Properties

1. Indefinite Integrals

   $$\int f(x)\,dx = F(x) + C$$

   Basic rules:

   • $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

   • $\int e^x dx = e^x + C$, $\int \sin x\,dx = -\cos x + C$, etc.

2. Integration Techniques

   • Substitution: Let $u = g(x)$, then $\int f'(g(x))g'(x)dx = \int f'(u)du$.

   • Integration by Parts:

     $$\int u\,dv = uv - \int v\,du$$

     Ideal for polynomial×log/trig.

3. Definite Integrals & Fundamental Theorem

   • $\int_a^b f(x)\,dx = F(b) - F(a)$

   • Properties:

     • $\int_a^a = 0$, $\int_a^b = -\int_b^a$

     • $\int_a^b [f+g] = \int_a^b f + \int_a^b g$

     • $\int_a^b f(x)dx = \int_a^c + \int_c^b$

4. Applications

   • Area under curves: $\int_a^b f(x) dx$ (above x-axis)

   • Area between curves: $\int_a^b [f(x) - g(x)] dx$

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📘 B. NCERT Exercise-Wise Solutions

• Ex 7.1: Indefinite integrals—apply standard formulas and substitution.

• Ex 7.2: Integration by parts—solve examples like $\int x\ln x\,dx$.

• Ex 7.3: Definite integrals—evaluate using FTC; properties problems.

• Ex 7.4: Area between curves—set up integrals and compute area.

Each solution includes:

1. Statement

2. Step-by-step integration method

3. Final result

4. Any substitution or limits used

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🏆 C. CBSE PYQs with Best Answers

1. 2023 (5 marks):

   $$\int_0^1 x\ln(1+x)\,dx$$

   Solution: Use part: $u=\ln(1+x)$, $dv=x\,dx$. Evaluate and simplify.

2. 2022 (3 marks):
   Find area between curves $y=x^2$ and $y=x$ from 0 to 1:

   $$\int_0^1 (x - x^2)\,dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{6}$$

3. 2021 (4 marks):

   $$\int x e^x\,dx$$

   Solution: Integration by parts with $u=x, dv=e^x dx$. Answer: $(x-1)e^x + C$.

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📌 #Tags:

#Class12Maths, #Integrals, #Calculus, #IntegrationTechniques, #CBSEPYQs, #NCERTSolutions, #AreaUnderCurve, #Chapter7Maths, #BoardExamPrep