Integration by Parts: ILATE Rule, Formula & Solved Examples (Class 12)

Integration by Parts: ILATE Rule, Formula & Solved Examples (Class 12)

When an integrand is a product of two different types of functions — like x·sin x, x²·eˣ, or log x on its own — no standard formula or simple substitution will crack it directly. This is where integration by parts comes in, a technique built directly on the product rule of differentiation, run in reverse. This guide explains the formula, the ILATE rule for choosing which function to integrate first, and walks through the exact worked examples that appear most often in CBSE Class 12 board exams.

The Integration by Parts Formula

Starting from the product rule of differentiation, d/dx(uv) = u(dv/dx) + v(du/dx), and integrating both sides, we arrive at:

∫u·(dv/dx) dx = uv − ∫v·(du/dx) dx

In words: ∫(first function)(second function) dx = (first function) × ∫(second function)dx − ∫[derivative of first function × ∫(second function)dx] dx

The entire technique hinges on one crucial decision: which of the two functions should be treated as the "first function" (the one you differentiate) and which as the "second function" (the one you integrate)?

The ILATE Rule: Choosing the First Function

ILATE is a memory device that ranks function types in the order they should be chosen as the first function:

• I — Inverse trigonometric functions (sin⁻¹x, tan⁻¹x, etc.)
• L — Logarithmic functions (log x)
• A — Algebraic functions (x, x², polynomials)
• T — Trigonometric functions (sin x, cos x)
• E — Exponential functions (eˣ, aˣ)

Whichever function type appears earliest in this list should be treated as the first function, since it generally becomes simpler when differentiated.

Worked Example: Choosing the Wrong Function First

Example 1: Evaluate ∫x cos x dx

Following ILATE, "algebraic" (x) comes before "trigonometric" (cos x), so x is the first function:

∫x cos x dx = x∫cos x dx − ∫[1·∫cos x dx]dx = x sin x − ∫sin x dx = x sin x + cos x + C

What happens if we choose incorrectly? If we instead pick cos x as the first function, we get x²/2 · cos x + ∫(x²/2)sin x dx — a more complicated integral than we started with, since the power of x increased instead of reducing to a constant. This demonstrates exactly why the ILATE order matters.

Worked Example: Logarithmic Functions

Example 2: Evaluate ∫log x dx

There's no obvious "second function" here, so we treat 1 as the second function (its integral is simply x):

∫(log x · 1) dx = log x · x − ∫(1/x)·x dx = x log x − ∫1 dx = x log x − x + C

Worked Example: Exponential Functions

Example 3: Evaluate ∫x eˣ dx

By ILATE, algebraic (x) comes before exponential (eˣ):

= x·eˣ − ∫1·eˣ dx = xeˣ − eˣ + C = eˣ(x − 1) + C

Worked Example: Inverse Trigonometric Functions

Example 4: Evaluate ∫tan⁻¹x dx

Similar to log x, treat 1 as the second function:

= x tan⁻¹x − ∫x · [1/(1+x²)] dx = x tan⁻¹x − (1/2)log(1+x²) + C

(The remaining integral is solved by substituting t = 1 + x², since its derivative 2x appears in the numerator.)

The Special Shortcut: ∫eˣ[f(x) + f′(x)]dx = eˣf(x) + C

One of the most exam-favorite patterns in this chapter is recognizing when an integrand secretly fits the form eˣ[f(x) + f′(x)]. When it does, the answer is immediately eˣf(x) + C — no lengthy integration by parts required.

Example 5: Evaluate ∫eˣ[1/x − 1/x²] dx

Let f(x) = 1/x, so f′(x) = −1/x². Then f(x) + f′(x) = 1/x − 1/x², which exactly matches the integrand.

= eˣ · (1/x) + C = eˣ/x + C

Spotting this pattern instantly — rather than attempting integration by parts the long way — can save several minutes in a timed exam.

Double Integration by Parts: The e^x sin x Type

Some integrals require applying integration by parts twice, and then solving algebraically for the original integral, since it reappears on both sides of the equation.

Example 6: Evaluate ∫eˣ sin x dx

Let I = ∫eˣ sin x dx. Applying parts once (with sin x as the first function):

I = −eˣ cos x + ∫eˣ cos x dx

Applying parts again to the remaining integral:

∫eˣ cos x dx = eˣ sin x − ∫eˣ sin x dx = eˣ sin x − I

Substituting back: I = −eˣ cos x + eˣ sin x − I, so 2I = eˣ(sin x − cos x), giving:

I = (eˣ/2)(sin x − cos x) + C

This "solve for I" technique is unique to this type of problem and is a favorite 5-mark board question.

Common Mistakes to Avoid

• Ignoring ILATE, which often makes the resulting integral harder, not easier.
• Forgetting to integrate the "second function" correctly before applying the formula.
• Missing the "−∫" sign, leading to an incorrect final answer.
• Not recognizing the eˣ[f(x)+f′(x)] shortcut, wasting time when a one-line answer was possible.

Suggested Internal Links

• Link to: Indefinite Integrals Class 12: Formulas & Solved Examples (for the base formulas used in every parts problem)
• Link to: Integration by Substitution Method (used within the "second function" steps of parts problems)
• Link to: Integration by Partial Fractions (the technique preceding this in most syllabi)
• Link to: Definite Integrals & Fundamental Theorem of Calculus (applying parts with limits)

Ready to Master Integration by Parts?

The ILATE rule is easy to state but takes real practice to apply instinctively under exam pressure. ChampionsPrep's AI-powered doubt resolution and topic-wise practice sets, at just ₹10 per use, let you drill this exact decision-making skill until it becomes second nature. Practice Integration by Parts on ChampionsPrep →

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