Explain integration by parts like I'm in class 12

Integration by parts is a technique to integrate products of two functions. The formula is derived from the product rule of differentiation. If you have two functions $u(x)$ and $v(x)$, then the product rule says $\frac{d}{dx}(u v) = u \frac{dv}{dx} + v \frac{du}{dx}$. Integrate both sides and rearrange to get the integration by parts formula: $$\int u \, dv = u v - \int v \, du$$.

Here, $u$ is the part you differentiate and $dv$ is the part you integrate. The trick is choosing $u$ wisely. Use the ILATE rule to pick $u$: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. Choose $u$ in that priority order, and the rest becomes $dv$.

For example, compute $\int x \cos x \, dx$. Here, $x$ is algebraic and $\cos x$ is trigonometric, so by ILATE, choose $u = x$ and $dv = \cos x \, dx$. Then $du = dx$ and $v = \sin x$. Plug into the formula: $$\int x \cos x \, dx = x \sin x - \int \sin x \, dx = x \sin x + \cos x + C$$.

Important: When you have to apply by parts twice (like $\int e^x \sin x \, dx$), don't forget to rearrange and solve for the original integral. Also, for definite integrals, evaluate $uv$ at the limits and subtract the integral of $v \, du$ with limits. Practice choosing $u$ with ILATE, and you'll crack any product integral in exams.

Remember: I L A T E – "I Love All Things Easier" – and always keep the $+C$ at the end for indefinite integrals.