explain integration by parts with formula

Integration by parts is the product rule of differentiation reversed. It helps integrate products of two functions. The formula is $$\int u\,dv = uv - \int v\,du$$. Here, u is the part you differentiate, and dv is the part you integrate. The goal is to choose u and dv so that the new integral $\int v\,du$ is simpler than the original.

Choose u using the ILATE rule (priority order): Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. So pick u as the function that comes first in ILATE; the rest becomes dv. For example, in $\int x\cos x\,dx$, u = $x$ (algebraic) and dv = $\cos x\,dx$. Then $du = dx$ and $v = \sin x$. Plug into formula: $$\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x + \cos x + C$$.

Key tricks: If you have $\ln x$ or $\sin^{-1}x$, treat them as u even if they seem odd—their derivative is simpler. For repeated integrals, apply parts twice (e.g., $\int x^2 e^x\,dx$). For cyclic integrals like $\int e^x\sin x\,dx$, use parts twice and solve algebraically for the integral.

Mnemonic: “u first, dv second, then uv minus the swap.” Always add the constant $C$ at the end. Practice choosing u wisely—a bad choice makes the integral harder. If you get stuck, try ILATE again or rearrange the integrand.