How do I decide which function to take as first in integration by parts (ILATE rule)?
GradeHunt_TeamGradeHunt Team · · CBSE 12th
Class 12 Maths integration by parts. My teacher said ILATE but in some questions it doesn't work. How should I choose u and v?
integration by partsilateintegralsclass 12 maths
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Integration by parts: integral of u.v dx = u.(integral of v dx) - integral of [ u'.(integral of v dx) ] dx, where u' = du/dx.
The ILATE rule helps choose u, the function you differentiate. Take u as the function that comes first in this order:
I - Inverse trigonometric (sin^-1 x, tan^-1 x)
L - Logarithmic (log x)
A - Algebraic (x, x^2)
T - Trigonometric (sin x, cos x)
E - Exponential (e^x)
Why it works: functions near the top become simpler when differentiated (log x becomes 1/x), and functions near the bottom are easy to integrate (e^x stays e^x).
Example: integral of x e^x dx
u = x (Algebraic), v = e^x (Exponential).
= x e^x - integral of e^x dx = x e^x - e^x + C.
When ILATE does not work directly:
1. Only one function, like integral of log x dx: take u = log x and v = 1. This gives x log x - x + C.
2. Integral of e^x sin x dx: apply by parts twice, and keep the same type of function as the first function both times (ILATE gives sin x first and then cos x; taking e^x first both times also works). The original integral appears again; solve for it like an equation to get (1/2) e^x (sin x - cos x) + C. If you switch the choice in the second step, you only get back to where you started.
3. Integral of e^x [f(x) + f'(x)] dx = e^x f(x) + C. Learn this standard result; it saves time in board exams.
4. Sometimes substitution must come first. In integral of x^3 e^(x^2) dx, e^(x^2) has no simple integral, so it cannot be the second function. Put x^2 = t, so 2x dx = dt and the integral becomes (1/2) integral of t e^t dt. Now use parts: (1/2)(t e^t - e^t) + C = (1/2) e^(x^2) (x^2 - 1) + C.
Exam tip: write "Taking x as the first function and e^x as the second function" before integrating, even in short answers. It makes your method clear, and CBSE marking schemes split the marks across the steps of a solution, not just the final answer.
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