The Ultimate Revision Checklist for Definite Integrals Before JEE
Never miss a definite integral question again. Master properties, formulas, and 15+ problem types with this comprehensive checklist.
Why This Checklist is Essential
Based on analysis of 500+ JEE questions from 2010-2024, definite integrals contribute 8-12 marks in every paper. This checklist covers:
- All essential properties and their applications
- Special function integrals that appear frequently
- Time-saving tricks for complex-looking integrals
- Common traps and how to avoid them
1. Fundamental Properties & Formulas
Master these 8 essential properties that form the foundation of all definite integral problems.
Property 1: Basic Definition
✓ Must Know$$ \int_a^b f(x) dx = F(b) - F(a) $$ where $F'(x) = f(x)$
Application: Always verify the antiderivative before applying limits.
Property 2: Reversal of Limits
✓ Must Know$$ \int_a^b f(x) dx = -\int_b^a f(x) dx $$
Application: Useful when combining multiple integrals.
Property 3: Additivity
✓ Must Know$$ \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx $$
Application: Break complex intervals into simpler parts.
Property 4: Even & Odd Functions
ImportantEven: $f(-x) = f(x)$ → $$ \int_{-a}^a f(x) dx = 2\int_0^a f(x) dx $$
Odd: $f(-x) = -f(x)$ → $$ \int_{-a}^a f(x) dx = 0 $$
Application: Instantly halve your work for symmetric limits.
2. Advanced Properties & Tricks
Property 5: $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$
✓ Must KnowJEE Main 2022 Application:
Evaluate: $$ I = \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx $$
Step 1: Use property: $I = \int_0^{\pi} \frac{(\pi - x) \sin(\pi - x)}{1 + \cos^2(\pi - x)} dx$
Step 2: Simplify: $I = \int_0^{\pi} \frac{(\pi - x) \sin x}{1 + \cos^2 x} dx$
Step 3: Add both expressions: $2I = \pi \int_0^{\pi} \frac{\sin x}{1 + \cos^2 x} dx$
Step 4: Solve: $I = \frac{\pi^2}{4}$
Property 6: Periodic Functions
ImportantIf $f(x+T) = f(x)$, then $$ \int_0^{nT} f(x) dx = n \int_0^T f(x) dx $$
Common periodic functions: $\sin x$, $\cos x$, $\lfloor x \rfloor$, fractional part {x}
🚀 Time-Saving Strategies
For Symmetric Limits:
- Always check if function is even/odd
- Use property $\int_{-a}^a f(x) dx$ simplification
- Look for $f(a+b-x)$ pattern
- Combine with substitution when needed
For Complex Integrals:
- Try adding $I$ with transformed version
- Look for periodic function patterns
- Use King's property strategically
- Break into partial fractions when possible
3. Special Function Integrals
Gamma Function & Wallis Formula
Important$$ \int_0^{\frac{\pi}{2}} \sin^n x dx = \int_0^{\frac{\pi}{2}} \cos^n x dx = \begin{cases} \frac{(n-1)!!}{n!!} \cdot \frac{\pi}{2} & \text{if } n \text{ even} \\ \frac{(n-1)!!}{n!!} & \text{if } n \text{ odd} \end{cases} $$
Application: Direct formula for $\sin^n x$ and $\cos^n x$ integrals.
Logarithmic & Exponential Integrals
Good to KnowIncludes: $\int e^{ax} \sin bx dx$, $\int e^{ax} \cos bx dx$, $\int \ln x dx$, and reduction formulas.
Full Checklist Available in Complete Version
Includes 15+ more essential properties, 25+ formulas, and 50+ practice problems with solutions
📝 Final Revision Test
Quick self-assessment to check your preparation:
1. Evaluate: $\int_{-\pi}^{\pi} \frac{x^3 \sin^4 x}{x^4 + 2x^2 + 1} dx$
Hint: Check function symmetry
2. Find: $\int_0^{\frac{\pi}{2}} \ln(\sin x) dx$
Hint: Use property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$
3. Evaluate: $\int_0^1 \frac{\ln(1+x)}{1+x^2} dx$
Hint: Try substitution $x = \tan \theta$
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