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:


🧠 A. Key Concepts & Properties

  1. Indefinite Integrals

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

    Basic rules:

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

    • exdx=ex+C\int e^x dx = e^x + C, sinxdx=cosx+C\int \sin x\,dx = -\cos x + C, etc.

  2. Integration Techniques

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

    • Integration by Parts:

      udv=uvvdu\int u\,dv = uv - \int v\,du

      Ideal for polynomial×log/trig.

  3. Definite Integrals & Fundamental Theorem

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

    • Properties:

      • aa=0\int_a^a = 0, ab=ba\int_a^b = -\int_b^a

      • ab[f+g]=abf+abg\int_a^b [f+g] = \int_a^b f + \int_a^b g

      • abf(x)dx=ac+cb\int_a^b f(x)dx = \int_a^c + \int_c^b

  4. Applications

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

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


📘 B. NCERT Exercise-Wise Solutions

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

  • Ex 7.2: Integration by parts—solve examples like xlnxdx\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


🏆 C. CBSE PYQs with Best Answers

  1. 2023 (5 marks):

    01xln(1+x)dx\int_0^1 x\ln(1+x)\,dx

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

  2. 2022 (3 marks):
    Find area between curves y=x2y=x^2 and y=xy=x from 0 to 1:

    01(xx2)dx=[x22x33]01=16\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):

    xexdx\int x e^x\,dx

    Solution: Integration by parts with u=x,dv=exdxu=x, dv=e^x dx. Answer: (x1)ex+C(x-1)e^x + C.


📌 #Tags:

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


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