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EAPCET Definite Integration – Complete Guide

🎯 EAPCET Definite Integration

Complete Master Guide with All Question Patterns

40
Symmetry & King’s Property
5
Major Topic Blocks
50+
Practice Problems
100
Topic Coverage

Block 1: Symmetry & King’s Property

⭐ The #1 High-Yield Topic – 40

Core Property
$$\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$$
💡 Recognition Pattern
When you see symmetric limits or expressions involving \(a+b-x\), immediately think of this property!
1. Fundamental Symmetry
Example (2023)
$$\int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} dx$$

Logic: \(a+b=3+6=9\). Apply King’s property.

Result: 1.5

2. Exponential Symmetry
Example (2024)
$$\int_{-\pi/15}^{\pi/15} \frac{\cos 5x}{1+e^{5x}} dx$$

Logic: Check \(f(x)+f(-x) = \cos 5x\) (constant).

Result: \(\frac{\sqrt{3}}{10}\)

3. Trig Reciprocal Pattern
Example (2024)
$$\int_{\pi/5}^{3\pi/10} \frac{dx}{\sec^2 x + (\tan^{2022} x – 1)(\sec^2 x – 1)}$$

Result: \(\pi/20\)

4. Logarithmic Symmetry
Example (2021)
$$\int_{-1}^1 \frac{\log(1+x)}{1+x^2} dx$$

Key: Add \(f(x)+f(-x)\) to eliminate the logarithm.

Result: \(\int_0^1 \frac{\log(1-x)}{1+x^2} dx\)

5. Fixed Value Pattern ⭐
If integrand = \(\frac{g(x)}{g(x)+g(a+b-x)}\), then answer = \(\frac{b-a}{2}\)
Example (2021)
$$\int_5^9 \frac{\log 3x^2}{\log 3x^2 + \log(588 – 84x + 3x^2)} dx$$

Shortcut: \(\frac{9-5}{2} = 2\)

Result: 2

6. Trig Power Symmetry
Example (2021)
$$\int_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx$$

Pattern: Symmetric form with \(\sin\) and \(\cos\).

Result: \(\pi/4\)

  • Variable Function Symmetry: \(\int_{-23}^{23} \frac{dx}{3 + f(x)}\) where \(f(x)f(-x) = 9\) 2023
    Result: 46/3
  • King’s Rule + Trig: \(\int_0^\pi \frac{x \cos^2 x}{1 + \sin x} dx\) 2023
    Result: \(\pi(\pi-2)/2\)
  • Substitution Symmetry: \(\int_0^{\pi/2} \frac{x \tan x \sec^2 x}{\tan^4 x + 1} dx\) 2023
    Result: \(\pi^2/16\)
  • King’s in Denominators: \(\int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx\) 2023
    Result: \(\frac{1}{\sqrt{2}}\log(\sqrt{2}+1) – \frac{\pi}{4}\)

Block 2: Special Functions

Modulus, GIF (Greatest Integer Function), Fractional Part

Breaking Points Methodology
📍 Key Strategy
  1. Modulus: Find zeros/roots → split integral at those points
  2. GIF [x]: Identify where function jumps (at integers)
  3. Fractional Part {x}: Use periodicity (period = 1)
1. Nested Modulus
Example (2022)
$$\int_0^4 ||x – 2| – x|dx$$

Breaking points: \(x=1, 2\)

Result: 6

2. Polynomial Modulus
Example (2023)
$$\int_0^3 |x^2 – 3x + 2| dx$$

Step 1: Factor: \((x-1)(x-2)\)

Step 2: Roots at \(x=1, 2\)

Step 3: Split: \([0,1] \cup [1,2] \cup [2,3]\)

Result: 11/6

3. Constant GIF Sum
Example (2023)
$$\int_0^{2\pi} [|\sin x| + |\cos x|] dx$$

Key insight: Value is always 1 in each interval.

Result: \(2\pi\)

4. GIF Transformation
Example (2023)
$$\int_{-2}^2 [2-x] dx$$

Substitution: Let \(u = 2-x\)

Result: 6

  • Fractional Part Periodicity: \(\int_0^2 \{x\} dx\) 2021
    Result: 1
  • Periodic GIF + Trig: \(\int_{-2}^2 \sin(x – [x]) dx\) 2021
    Result: \(4(1 – \cos 1)\)
  • Squared GIF: \(\int_1^2 [x^2] dx\) 2025
    Result: \(3 – \sqrt{2} – \sqrt{3}\)
  • Logarithmic Modulus: \(\int_{1/2}^2 |\log_{10} x| dx\) 2023
    Result: \(\frac{1}{\ln 10}[0.5 + \ln 2]\)
  • Odd Function Modulus: \(\int_{-1}^1 x|x| dx\) 2023
    Result: 0
  • Square Root Modulus: \(\int_0^{25\pi} \sqrt{|\cos x – \cos^3 x|} dx\) 2024
    Result scaled to k=4

Block 3: Wallis & Beta Reduction Formulas

Shortcuts for higher power trigonometric and algebraic integrals

Wallis Formula
$$\int_0^{\pi/2} \sin^m x \cos^n x \, dx = \frac{(m-1)!! \cdot (n-1)!!}{(m+n)!!} \times \begin{cases} \frac{\pi}{2} & \text{both even} \\ 1 & \text{one odd} \end{cases}$$
🔢 Double Factorial Reminder

\(n!! = n \times (n-2) \times (n-4) \times \cdots\)

Example: \(6!! = 6 \times 4 \times 2 = 48\)

Example: \(5!! = 5 \times 3 \times 1 = 15\)

Beta Formula
$$\int_0^a x^m(a-x)^n dx = a^{m+n+1} \cdot \frac{m! \cdot n!}{(m+n+1)!}$$
1. Standard Wallis
Example (2023)
$$\int_0^{\pi/2} \sin^6 x \cos^4 x \, dx$$

Solution: \(m=6, n=4\) (both even)

\(\frac{5!! \cdot 3!!}{10!!} \cdot \frac{\pi}{2} = \frac{15 \cdot 3}{3840} \cdot \frac{\pi}{2}\)

Result: \(3\pi/512\)

2. Even Power Wallis
Example (2025)
$$\int_{-2\pi}^{2\pi} \sin^4 x \cos^6 x \, dx$$

Note: Even function, so double the result from \(0\) to \(2\pi\)

Result: \(3\pi/128\)

3. Beta Substitution
Example (2023)
$$\int_0^2 x^3(2-x)^4 dx$$

Direct Beta: \(a=2, m=3, n=4\)

\(2^8 \cdot \frac{3! \cdot 4!}{8!}\)

Result: 256/35

4. Radical Beta Form
Example (2023)
$$\int_0^2 x^{5/2} \sqrt{2-x} \, dx$$

Rewrite: \(x^{5/2}(2-x)^{1/2}\)

Use Beta with fractional powers

Result: \(32\sqrt{2}/63\)

  • Trig Power Wallis: \(\int_0^{\pi/2} \sin^4 \theta \cos^3 \theta \, d\theta\) 2022
    Result: 2/35
  • Reduction Challenge: \(\int_0^{\pi/2} \tan^{14}(\frac{x}{2})dx\) 2025
    Alternating sum form
  • Algebraic Wallis: \(\int_{-2}^2 x^4(4-x^2)^{7/2} dx\) 2024
    Result: \(\frac{3\pi}{64} \cdot 2^{12}\)
  • Even Part Wallis: \(\int_{-\pi/2}^{\pi/2} \sin^2 x \cos^2 x (\sin x + \cos x) dx\) 2023
    Result: \(\pi/16\)
  • Geometry + Wallis: \(\int_2^5 \sqrt{\frac{5-x}{x-2}} dx\) 2023
    Result: \(3\pi/2\)
  • Wallis Sine Power: \(\int_0^\pi \sqrt{1 – \cos 4x} \, dx\) 2024
    Result: \(2\sqrt{2}\)

Block 4: Riemann Sums (Limit as a Sum)

Converting discrete summation to continuous integration

Standard Conversion Formula
$$\lim_{n \to \infty} \frac{1}{n}\sum_{r=1}^{n} f\left(\frac{r}{n}\right) = \int_0^1 f(x)dx$$
🎯 Step-by-Step Process
  1. Factor out \(\frac{1}{n}\) from the entire expression
  2. Identify the pattern \(f(r/n)\)
  3. Replace \(\frac{r}{n} \to x\) and \(\frac{1}{n} \to dx\)
  4. Adjust limits: \(r=1\) to \(n\) becomes \(x=0\) to \(1\) (or \(x=0\) to \(p\) if needed)
1. Exponential Riemann
Example (2025)
$\lim_{n \to \infty} \frac{1}{n^2} [e^{1/n} + 2e^{2/n} + \dots + 2ne^2]$

Solution: Factor \(\frac{1}{n}\) and recognize pattern

Result: \(e^2+1\)

2. Logarithmic Riemann
Example (2021)
$\lim_{n \to \infty} \frac{1}{n} \log\left(\frac{(2n)!}{n^n \cdot n!}\right)$

Result: \(\log(4/e)\)

3. Algebraic Riemann
Example (2022)
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{np} \left(\left(\frac{r}{n}\right)^2 + 2\right)$

Convert to: \(\int_0^p (x^2 + 2)dx\)

Result: \(\frac{p^3}{3} + 2p\)

4. Trig Riemann Sum
Example (2023)
$\lim_{n \to \infty} \left[\frac{1}{n^2} \sec^2 \frac{1}{n^2} + \dots + \frac{1}{n} \sec^2 1\right]$

Convert to: \(\int_0^1 \sec^2 x \, dx\)

Result: \(\frac{1}{2} \tan 1\)

  • Log Sum Shift: Find \(f(x)\) if limit matches \(\int_1^2 f(x)dx\) 2021
    Result: \(f(x) = \log x\)
  • Power Limit: \(\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n \frac{r^k}{n^k}\) form Multiple Years
  • Inverse Trig Riemann: \(\lim_{n \to \infty} \sum_{r=1}^n \frac{n}{n^2+r^2}\) 2022
  • Exponential Base: \(\lim_{n \to \infty} \left( \frac{(n+1)(n+2)\dots(2n)}{n^n} \right)^{1/n}\) 2023

Block 5: Substitution & Algebraic Identity

Complex substitutions and specific definite integral properties

1. Exponential Log
Example (2022)
$\exp\left(\int_0^1 2x \log(1+x^2)dx\right)$

Use: Integration by parts or substitution

Result: \(4/e\)

2. Integer Root Equation
Example (2023)

If \(\int_0^3 (3x^2 – 4x + 2) dx = k\), solve \(3x^2 – 4x + 2 = 3k/5\)

Result: \(x = -1\)

3. IBP Coefficient Sum
Example (2023)

If \(\int x^4(\log x)^3 dx = \dots\), find \(A+B+C+5D\)

Result: 0

4. Symmetric Radical
Example (2021)
$\int_0^\pi \sqrt{1 + 4\sin^2 \frac{x}{2} + 4\sin \frac{x}{2}} \, dx$

Simplify: Complete the square inside

Result: \(\pi+4\)

  • Trig Rational Sub: \(\int_0^{\pi/4} \frac{\sec x}{3\cos x + 4\sin x} dx\) 2025
    Logarithmic form
  • Inverse Trig Identity: \(\int_0^{\pi/2} \frac{\pi \sin x}{1 + \cos^2 x} dx\) 2021
    Result: \(\pi^2/4\)
  • Geometric Radical: \(\int_1^2 x\sqrt{4-x^2} \, dx\) 2022
    Result: \(\sqrt{3}\)
  • Fractional Integral: \(\int_8^{18} \frac{1}{(x+2)\sqrt{x-3}} dx\) 2025
  • Trig Split: \(\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}\) 2021
    Result: 2

📚 Complete Study Strategy

Week-by-week plan for mastering definite integration

🎯 Week 1: Block 1 – Symmetry & King’s Property (40
  • Day 1-2: Master the core property formula and practice 10 fundamental symmetry problems
  • Day 3-4: Focus on exponential and logarithmic symmetry patterns
  • Day 5-6: Practice fixed value patterns and trig power symmetry
  • Day 7: Mixed practice – solve 20 problems from all Block 1 patterns

📊 Week 2: Blocks 2 & 3 (35
  • Day 1-3: Block 2 – Modulus, GIF, and fractional part functions
  • Day 4-5: Memorize Wallis and Beta formulas with double factorial practice
  • Day 6-7: Apply formulas to 15+ problems from both blocks

🔢 Week 3: Blocks 4 & 5 + Integration (25
  • Day 1-2: Riemann sums – master the limit-to-integral conversion
  • Day 3-4: Substitution techniques and algebraic identities
  • Day 5-7: Mixed practice from ALL blocks with timed tests

🎓 Exam Day Checklist
Quick Recognition Guide
  1. See symmetric limits? → Try King’s property first (40
  2. See |…| or […]? → Find breaking points
  3. See sin^m x cos^n x? → Check if Wallis applies
  4. See limit of sum? → Convert to Riemann integral
  5. See x^m(a-x)^n? → Beta formula shortcut
⚡ Time-Saving Tips
  • Fixed Value Pattern: If integrand = \(\frac{g(x)}{g(x)+g(a+b-x)}\), instantly write \((b-a)/2\)
  • Even Functions: \(\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx\) if \(f(-x)=f(x)\)
  • Odd Functions: \(\int_{-a}^a f(x)dx = 0\) if \(f(-x)=-f(x)\)
  • Wallis Quick Check: Both powers even? Multiply by \(\pi/2\). One odd? No \(\pi\).
📈 Practice Distribution
Total Problems to Solve: 100+
Block 1 (Symmetry): 40 problems (40 Block 2 (Special Functions): 20 problems (20 Block 3 (Wallis & Beta): 15 problems (15 Block 4 (Riemann): 15 problems (15 Block 5 (Substitution): 10 problems (10

💪 Final Week Strategy

  • Solve 5 full-length mock tests with all question types
  • Review mistakes and identify weak patterns
  • Create a formula sheet with all key formulas
  • Practice mental calculation for double factorials
  • Time yourself: aim for 2-3 minutes per problem

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