Explain integration by parts like I'm in class 12

Integration by parts is a technique to integrate a product of two functions. The rule comes from the product rule of differentiation. If you have $\int u\,dv$, you choose one part as $u$ (to differentiate) and the other as $dv$ (to integrate). The formula is: $$\int u\,dv = uv - \int v\,du$$. This formula is your best friend for questions like $\int x e^x dx$ or $\int x \sin x dx$.

The trick is choosing $u$ correctly. Use the ILATE rule to pick $u$: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. Whichever comes first in ILATE becomes $u$. For example, in $\int x e^x dx$, $x$ is algebraic and $e^x$ is exponential, so $u = x$ and $dv = e^x dx$. Then $du = dx$ and $v = e^x$. Plug into formula: $\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C = e^x(x-1) + C$. See how easy it is?

For integrals where one function doesn't become simpler on differentiation, like $\int \log x dx$, treat it as $\int 1 \cdot \log x dx$. Here $\log x$ is logarithmic (comes before algebraic $1$), so $u = \log x$, $dv = dx$. Then $du = \frac{1}{x}dx$ and $v = x$. So $\int \log x dx = x\log x - \int x \cdot \frac{1}{x}dx = x\log x - x + C$.

In board exams, you'll often need to apply the rule twice, like for $\int x^2 e^x dx$. Keep applying until one integral is simple. Also remember, when you have a single function like $\int \arcsin x dx$, add a $1$ as a factor. Practice choosing $u$ fast – that's half the battle. And always add $+C$ at the end. With regular practice, integration by parts becomes a scoring topic.

Remember, if you ever get stuck, just write the formula and match $u, dv, du, v$ – you'll crack it.