The direct answer
Integration by Parts — The LIATE Rule
Formula: ∫u dv = uv − ∫v du
The key is choosing u and dv correctly. Use the LIATE priority order:
Logarithmic (ln x) > Inverse trig (sin⁻¹x) > Algebraic (x, x²) > Trigonometric (sin x, cos x) > Exponential (eˣ)
Choose the function that appears first in LIATE as u (the one to differentiate). The rest becomes dv (the one to integrate).
🧠 Memory trick: "LIATE" — "Logs, Inverse, Algebra, Trig, Exponents". Think: "Lions In Africa Travel Everywhere".
Common patterns:
∫x·eˣdx → u=x (Algebraic before Exponential)
∫x·sin x dx → u=x (Algebraic before Trigonometric)
∫ln x dx → u=ln x (Logarithmic first)
∫x·ln x dx → u=ln x (Logarithmic before Algebraic)
∫sin⁻¹x dx → u=sin⁻¹x (Inverse trig first)
Substitution Method — Spot the Derivative
Core idea: If you see a function and its derivative in the integrand, substitute the function.
Step 1: Identify the inner function g(x) and set u = g(x)
Step 2: Find du = g'(x)dx
Step 3: Rewrite the integral in terms of u
Step 4: Integrate with respect to u, then substitute back
| Integrand Pattern | Substitution | Result |
|---|---|---|
| ∫f(g(x))·g'(x)dx | u = g(x) | ∫f(u)du — the general rule |
| ∫tan x dx | u = cos x | −ln|cos x| = ln|sec x| |
| ∫cot x dx | u = sin x | ln|sin x| |
| ∫x·√(x²+1) dx | u = x²+1 | (1/3)(x²+1)^(3/2) |
| ∫eˣ·f(eˣ) dx | u = eˣ | ∫f(u)du |
🧠 Memory trick: "If you see a function and its derivative, substitute." The derivative doesn't have to be exact — you can multiply/divide by constants to adjust.
Partial Fractions — Decompose and Integrate
When to use: Integrating rational functions P(x)/Q(x) where the denominator can be factored.
Step 1: Ensure degree of P(x) < degree of Q(x). If not, do polynomial division first.
Step 2: Factor the denominator completely.
Step 3: Decompose into simpler fractions.
Step 4: Find constants using the cover-up rule or by equating coefficients.
| Denominator Factor | Decomposition Form |
|---|---|
| Linear (x − a) | A/(x − a) |
| Repeated linear (x − a)ⁿ | A₁/(x−a) + A₂/(x−a)² + ... + Aₙ/(x−a)ⁿ |
| Quadratic (x² + px + q) | (Ax + B)/(x² + px + q) |
| Repeated quadratic (x² + px + q)ⁿ | Sum of (Aₖx + Bₖ)/(x² + px + q)ᵏ for k = 1 to n |
🧠 Memory trick: "Cover-up rule" — to find A for (x−a) factor, cover (x−a) in the original fraction and evaluate at x = a. Fastest method for linear factors.
Trigonometric Integrals — The Essential Identities
Key identities for JEE integration:
sin²x = (1 − cos 2x)/2
cos²x = (1 + cos 2x)/2
sin 3x = 3 sin x − 4 sin³x
cos 3x = 4 cos³x − 3 cos x
sin x cos x = (1/2) sin 2x
1 + tan²x = sec²x
| Integral Type | Method | Result |
|---|---|---|
| ∫sin²x dx | Use cos 2x identity | x/2 − sin 2x/4 + C |
| ∫cos²x dx | Use cos 2x identity | x/2 + sin 2x/4 + C |
| ∫sin³x dx | sin³x = sin x(1−cos²x) | −cos x + cos³x/3 + C |
| ∫sec³x dx | Integration by parts | (1/2)(sec x tan x + ln|sec x + tan x|) + C |
| ∫sinⁿx cosᵐx dx | n odd → substitute cos x; m odd → substitute sin x | Reduce using identities |
🧠 Memory trick: "Odd → swap, Even → reduce." If one power is odd, substitute the other function. If both are even, use double-angle identities to reduce.
Definite Integral Shortcuts — King's Rule & Symmetry
King's Rule: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
Application: If f(x) + f(a−x) = constant, the integral equals (constant × a)/2.
Odd/Even functions: For limits [−a, a]: odd function → 0, even function → 2×∫₀ᵃ
Periodic functions: ∫₀ⁿᵃ f(x)dx = n × ∫₀ᵃ for periodic function with period a.
🧠 Memory trick: "King's Rule King's Rule — flip the bounds, same result." When you see ∫₀ᵃ, always check if f(x) + f(a−x) simplifies. This is the fastest way to solve definite integrals in JEE.
Classic examples:
∫₀^(π/2) sin x/(sin x + cos x) dx = π/4
∫₀^(π/2) ln(sin x) dx = −(π/2)ln 2
∫₋₁¹ x³ dx = 0 (odd function)
∫₋₁¹ x² dx = 2∫₀¹ x² dx = 2/3 (even function)
JEE PYQ Patterns on Integration
| Question Type | Frequency | How to Approach |
|---|---|---|
| Integration by parts | Very High | Use LIATE rule, apply formula, repeat if needed |
| Definite integral with King's rule | Very High | Check f(x) + f(a−x) = constant |
| Substitution method | High | Spot function + derivative pattern |
| Partial fractions | High | Factor denominator, decompose, integrate |
| Trigonometric identities | Medium-High | Apply double-angle/half-angle identities |
| eˣ(f(x) + f'(x)) form | Medium | Result = eˣ·f(x) + C — spot the pattern |
Common JEE traps to avoid:
❌ Forgetting the + C constant in indefinite integrals
❌ Choosing wrong u in by parts (should use LIATE)
❌ Not recognizing eˣ(f(x) + f'(x)) pattern
❌ Forgetting King's rule when limits are 0 to a
❌ Not checking if function is odd/even for symmetric limits
❌ Making substitution but not changing limits in definite integrals
Frequently Asked Questions
What is integration by parts?
Integration by parts uses the formula ∫u dv = uv − ∫v du. Choose 'u' as the function that simplifies when differentiated (LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). Choose 'dv' as the function that's easy to integrate. This is the most important technique for JEE.
How do I choose u and dv in integration by parts?
Use the LIATE rule: Choose 'u' in this order — Logarithmic > Inverse trigonometric > Algebraic > Trigonometric > Exponential. The function that appears first in LIATE becomes 'u'. The remaining function becomes 'dv'. This ensures the integral simplifies on repeated application.
When do I use partial fractions in integration?
Use partial fractions when integrating rational functions (polynomial divided by polynomial) where the denominator can be factored. Decompose the fraction into simpler fractions, then integrate each term. This is essential for JEE questions involving rational functions.
What is the King's rule in definite integration?
The King's rule states: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx. This is used to evaluate definite integrals by symmetry. It's especially useful when f(x) + f(a−x) simplifies to a constant. A classic JEE shortcut for definite integrals.
What are the most common integration questions in JEE?
JEE frequently asks: (1) Integration by parts with log/trig functions, (2) Substitution with trigonometric identities, (3) Partial fractions with quadratic denominators, (4) Definite integrals using King's rule, (5) Properties of definite integrals (odd/even functions), (6) Integration of exponential times trigonometric functions.
How do I integrate eˣ times trigonometric functions?
Integrals of the form ∫eˣ(f(x) + f'(x))dx = eˣ·f(x) + C. For ∫eˣ sin(x)dx or ∫eˣ cos(x)dx, use integration by parts twice — the integral repeats, allowing you to solve algebraically. This is a guaranteed JEE question pattern.
What formulas should I memorize for integration?
Essential formulas: ∫xⁿdx = xⁿ⁺¹/(n+1), ∫1/x dx = ln|x|, ∫eˣdx = eˣ, ∫sin x dx = −cos x, ∫cos x dx = sin x, ∫sec²x dx = tan x, ∫1/(1+x²)dx = tan⁻¹x, ∫1/√(1−x²)dx = sin⁻¹x, ∫sec x tan x dx = sec x, ∫cosec²x dx = −cot x. Also memorize ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx.
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Master integration techniques, then move to limits, continuity, and differential equations.
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