On this page
Integration by parts is the method most students know as a formula and still get wrong in tests. The formula is rarely the problem. The problem is the first move: which function you differentiate and which you integrate. Choose badly and the new integral is harder than the one you started with. This post covers the rule of thumb for that choice, four worked examples, the cases where the rule breaks, and a routine to make the choice automatic.
What does integration by parts actually say?
NCERT's Class 12 chapter on Integrals derives it from the product rule. For a first function f and a second function g:
∫ f(x) g(x) dx = f(x) ∫ g(x) dx − ∫ [f′(x) ∫ g(x) dx] dx
In words: keep the first function, integrate the second, then subtract the integral of (derivative of the first) times (integral of the second). The method is listed by name, "by parts", in the integral calculus unit of the official JEE (Main) 2026 syllabus, alongside substitution and partial fractions.
The formula only helps if f′(x) ∫ g(x) dx is simpler than f(x) g(x). That is the whole game.
What is LIATE and why does it work?
LIATE is an ordering for picking the first function, usually traced to a short note by Herbert Kasube in The American Mathematical Monthly (1983). Take as the first function whichever comes earliest in this list:
Logarithmic: log x
Inverse trigonometric: tan⁻¹ x, sin⁻¹ x
Algebraic: x, x², polynomials
Trigonometric: sin x, cos x
Exponential: eˣ, e²ˣ
The logic is simple. Functions near the front are hard to integrate but become simpler when differentiated (log x becomes 1/x). Functions near the end are easy to integrate again and again. NCERT gives the same advice without the acronym: take a polynomial as the first function, except when the other function is inverse trigonometric or logarithmic, in which case take that one first. Many Indian books write the same list as ILATE, swapping the first two; for the products you meet in school the two orders rarely disagree.
Four worked examples
1. ∫ x e²ˣ dx. Algebraic beats exponential, so f = x and g = e²ˣ. Then ∫ g dx = e²ˣ/2 and f′ = 1.
∫ x e²ˣ dx = x e²ˣ/2 − ∫ e²ˣ/2 dx = x e²ˣ/2 − e²ˣ/4 + C
Check by differentiating: e²ˣ/2 + x e²ˣ − e²ˣ/2 = x e²ˣ. Correct.
2. ∫ x log x dx. Logarithmic beats algebraic, so f = log x and g = x. Then ∫ g dx = x²/2 and f′ = 1/x.
∫ x log x dx = (x²/2) log x − ∫ (x²/2)(1/x) dx = (x²/2) log x − x²/4 + C
If you had chosen f = x, you would need ∫ log x dx inside the formula, which is itself a by-parts problem. LIATE saves a step.
3. ∫ x tan⁻¹ x dx. Inverse trigonometric beats algebraic, so f = tan⁻¹ x and g = x.
∫ x tan⁻¹ x dx = (x²/2) tan⁻¹ x − (1/2) ∫ x²/(1 + x²) dx
Write x²/(1 + x²) = 1 − 1/(1 + x²), so the last integral is x − tan⁻¹ x. The answer is ((x² + 1)/2) tan⁻¹ x − x/2 + C. Differentiating gives x tan⁻¹ x + 1/2 − 1/2, which checks out.
4. A definite integral: ∫₀¹ x eˣ dx. With f = x, the antiderivative is x eˣ − eˣ. At x = 1 it is e − e = 0. At x = 0 it is 0 − 1 = −1. So the integral is 0 − (−1) = 1. A positive answer makes sense, since x eˣ is positive on (0, 1).
When does LIATE not work?
Treat LIATE as a first guess, not a law. Three cases come up often.
A hidden substitution. For ∫ x³ e^(x²) dx, LIATE says f = x³, but e^(x²) has no elementary integral. Substitute t = x² first, so dt = 2x dx and the integral becomes (1/2) ∫ t eᵗ dt. Now by parts gives (1/2)(t eᵗ − eᵗ), that is, (1/2) e^(x²)(x² − 1) + C.
Cyclic integrals. For ∫ eˣ sin x dx, both functions keep coming back. NCERT works this one: apply by parts twice, the original integral reappears, and you solve for it as an unknown. The answer is eˣ(sin x − cos x)/2 + C.
A single function. For ∫ log x dx or ∫ tan⁻¹ x dx, take the constant function 1 as the second function. NCERT does exactly this for log x, giving x log x − x + C.
NCERT also proves the useful special case ∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C. If you spot an eˣ multiplied by a function plus its derivative, you can skip by parts entirely.
Common mistakes
Reversing the roles. Taking f = eˣ in ∫ x eˣ dx gives a new integral with x²/2 in it, which is worse. If the new integral looks harder, stop and swap.
Dropping a sign. The minus in front of the second integral is lost more often than any other symbol. Write it before you fill in anything else.
Adding a constant to ∫ g dx. NCERT shows that a constant added at this step cancels out. Add C once, at the end.
Not checking. Differentiating your answer takes thirty seconds and catches most errors.
A two-week practice routine
Days 1 to 3: for each of 15 products, write only the first function and a one-line reason. Do not integrate yet.
Days 4 to 7: solve eight by-parts integrals a day, then differentiate each answer to check it.
Days 8 to 10: mix by parts with substitution and partial fractions so you must choose the method too. In a study of 126 seventh-grade maths students, Rohrer, Dedrick and Stershic (2015) found that interleaved practice, where problem types are mixed, produced higher scores than practising one type at a time, on a test one day later and on another 30 days later. Our integration practice plan and the guide to partial fractions give sets to draw from.
Days 11 to 14: five cyclic or substitution-first problems a day, then a timed set of ten.
If you keep choosing the wrong first function and cannot see why, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can compare its first move with yours.
Tonight, take ∫ x² eˣ dx, name the first function, solve it, and differentiate your answer to check.
