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Calculus Mastery Reading Time: 15 min Essential for JEE

Integration by Parts: The ILATE Rule for JEE Success

Master the most powerful integration technique with ILATE priority rule. Solve complex JEE integrals systematically.

2-3
Questions per Paper
85%
JEE Relevance
5
Key Applications
12+
Solved Examples

Why Integration by Parts is Crucial for JEE

Integration by Parts appears in 2-3 questions in every JEE paper, making it one of the most frequently tested calculus topics. Mastering this technique with the ILATE rule can help you secure 8-12 marks easily.

🎯 JEE Examination Pattern

  • JEE Main: 1-2 direct questions on integration by parts
  • JEE Advanced: Complex applications in definite integrals
  • Common combinations: With trigonometric, exponential, and logarithmic functions
  • Time-saving: ILATE rule helps solve in 60-90 seconds

🎯 Quick Navigation

Basic Formula ILATE Rule Solved Examples JEE Applications

1. The Integration by Parts Formula

Fundamental Formula

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

Where u and v are functions of x

Understanding the Formula

The formula is derived from the product rule of differentiation:

$$ \frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx} $$

Integrating both sides:

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

Rearranging gives us the integration by parts formula.

💡 Memory Aid

"Ultra Violet - Very Unusual" helps remember:

∫ u dv = uv - ∫ v du

2. The ILATE Rule for Choosing 'u'

ILATE Priority Order

I
Inverse Trigonometric
L
Logarithmic
A
Algebraic
T
Trigonometric
E
Exponential

How ILATE Works

When faced with product of two functions, choose 'u' according to ILATE priority:

  • Inverse Trigonometric functions have highest priority
  • Logarithmic functions come next
  • Algebraic functions (polynomials, roots) follow
  • Trigonometric functions come after algebraic
  • Exponential functions have the lowest priority

Example: Choosing 'u' with ILATE

For ∫ x · eˣ dx:

Step 1: Identify functions: Algebraic (x) and Exponential (eˣ)

Step 2: According to ILATE: Algebraic > Exponential

Step 3: Choose u = x (Algebraic), dv = eˣ dx

This choice simplifies the integral effectively.

3. Step-by-Step Solving Procedure

5-Step Method for Integration by Parts

Step 1: Identify Functions

Break the integrand into product of two functions

Step 2: Apply ILATE Rule

Choose 'u' according to ILATE priority order

Step 3: Differentiate and Integrate

Find du (differentiate u) and v (integrate dv)

Step 4: Apply Formula

Substitute into ∫ u dv = uv - ∫ v du

Step 5: Simplify

Solve the remaining integral and simplify the expression

4. Solved Examples with ILATE Rule

Example 1 JEE Main 2023 Pattern

Algebraic × Exponential

Evaluate ∫ x · eˣ dx

Solution:

Step 1: Identify functions: Algebraic (x) and Exponential (eˣ)

Step 2: ILATE: Algebraic > Exponential, so u = x, dv = eˣ dx

Step 3: du = dx, v = ∫ eˣ dx = eˣ

Step 4: Apply formula:

∫ x · eˣ dx = x · eˣ - ∫ eˣ dx

Step 5: Simplify:

= x eˣ - eˣ + C = eˣ(x - 1) + C
Example 2 JEE Advanced 2022 Pattern

Logarithmic × Algebraic

Evaluate ∫ ln x dx

Solution:

Step 1: Write as ∫ 1 · ln x dx

Step 2: ILATE: Logarithmic > Algebraic, so u = ln x, dv = dx

Step 3: du = (1/x) dx, v = x

Step 4: Apply formula:

∫ ln x dx = x ln x - ∫ x · (1/x) dx

Step 5: Simplify:

= x ln x - ∫ dx = x ln x - x + C = x(ln x - 1) + C
Example 3 JEE Main 2021

Algebraic × Trigonometric

Evaluate ∫ x sin x dx

Solution:

Step 1: Identify functions: Algebraic (x) and Trigonometric (sin x)

Step 2: ILATE: Algebraic > Trigonometric, so u = x, dv = sin x dx

Step 3: du = dx, v = ∫ sin x dx = -cos x

Step 4: Apply formula:

∫ x sin x dx = x(-cos x) - ∫ (-cos x) dx

Step 5: Simplify:

= -x cos x + ∫ cos x dx = -x cos x + sin x + C

5. Special Cases & JEE Applications

Cyclic Integration by Parts

Some integrals require applying integration by parts twice, bringing you back to the original integral.

Example: ∫ eˣ sin x dx

Let I = ∫ eˣ sin x dx

First application: u = sin x, dv = eˣ dx

I = eˣ sin x - ∫ eˣ cos x dx

Second application on ∫ eˣ cos x dx: u = cos x, dv = eˣ dx

I = eˣ sin x - [eˣ cos x + ∫ eˣ sin x dx]

I = eˣ sin x - eˣ cos x - I

2I = eˣ(sin x - cos x)

I = (eˣ/2)(sin x - cos x) + C

Definite Integrals with Integration by Parts

For definite integrals ∫ab u dv, the formula becomes:

$$ \int_a^b u \, dv = [uv]_a^b - \int_a^b v \, du $$

Example: ∫0π x sin x dx

u = x, dv = sin x dx

du = dx, v = -cos x

∫0π x sin x dx = [-x cos x]0π + ∫0π cos x dx

= [-π cos π - 0] + [sin x]0π

= [-π(-1)] + [0 - 0] = π

6. Common Mistakes to Avoid

❌ Wrong 'u' Selection

  • Choosing exponential over logarithmic
  • Not following ILATE priority
  • Making integral more complicated

❌ Integration Errors

  • Wrong integration of dv
  • Forgetting constant of integration
  • Sign errors in formula application

✅ Prevention Strategy

  • Always write ILATE and follow it strictly
  • Double-check integration of dv
  • Practice with timer to build speed
  • Verify answers by differentiation

7. Practice Problems for JEE

Try These JEE-Level Problems

1. ∫ x² eˣ dx

Hint: Apply integration by parts twice

2. ∫ x ln x dx

Hint: Logarithmic function has priority

3. ∫ eˣ cos x dx

Hint: Cyclic integration by parts

4. ∫01 x eˣ dx

Hint: Definite integral application

Time yourself: Try to solve each problem within 2 minutes!

8. Quick Revision Checklist

📝 Must-Remember Formulas

  • ∫ u dv = uv - ∫ v du
  • ILATE: I → L → A → T → E
  • Definite: ∫ab u dv = [uv]ab - ∫ab v du
  • Cyclic: Remember the pattern

🎯 Exam Strategy

  • Always apply ILATE rule first
  • Check if integral simplifies
  • Watch for cyclic patterns
  • Verify by differentiation

Mastered Integration by Parts?

Continue your calculus journey with these advanced topics

Definite Integrals → All Integration Methods