Q.$e^{x}\left(\dfrac{x-1}{2x^{2}}\right)$

Integration by Parts

The idea: reverse the product rule

Some integrands are a product of two very different functions — $x e^x$, $x\cos x$, $\log x$, $x\sin^{-1}x$ — where substitution gets you nowhere. Integration by parts is the tool for these. It comes straight from reversing the product rule for differentiation.

Starting from $\dfrac{d}{dx}(uv)=u\,v' + u'\,v$ and integrating both sides gives the working formula:

In words: integral of (first $\times$ derivative-of-second) $=$ first $\times$ integral-of-second $-$ integral of (derivative-of-first $\times$ integral-of-second).

Choosing $u$: the ILATE rule

The whole game is picking which factor is $u$ (to differentiate) and which is $dv$ (to integrate). Pick $u$ by ILATE — the first type that appears:

• Inverse trig ($\sin^{-1}x$), Logarithmic ($\log x$), Algebraic ($x^2$), Trigonometric ($\sin x$), Exponential ($e^x$).

Whatever comes first in ILATE becomes $u$; the rest is $dv$. This makes the new integral $\int v\,du$ simpler than the one you started with.

Worked idea

For $\int x e^x\,dx$: algebraic before exponential, so $u=x$, $dv=e^x dx$. Then $du=dx$, $v=e^x$:

Integration by Parts is one of the most tested methods in the NCERT Class 12 Mathematics chapter on Integrals, and "integration by parts formula ILATE rule" along with "integration by parts class 12 important questions" are among the top searches for students preparing for CBSE board exams and JEE Main calculus. The same ILATE-based technique extends naturally into JEE Advanced integral calculus problems built on this NCERT Class 12 foundation.