explain integration by parts with formula

Integration by parts is a technique to integrate products of two functions. It comes from the product rule of differentiation. If you have two functions $u(x)$ and $v(x)$, then the product rule says: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$. Integrate both sides with respect to $x$: $\int \frac{d}{dx}(uv)\,dx = \int u\frac{dv}{dx}\,dx + \int v\frac{du}{dx}\,dx$. The left side is just $uv$, so we get $uv = \int u\,dv + \int v\,du$. Rearranging gives the integration by parts formula: $$\int u\,dv = uv - \int v\,du$$. Here, choose $u$ as the function that simplifies when differentiated, and choose $dv$ as the rest that you can integrate. The ILATE rule helps pick $u$: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. So prefer $u$ in that order. For example, integrate $\int x e^x\,dx$. Let $u = x$ (algebraic, since log/ inverse come first but none here), $dv = e^x\,dx$. Then $du = dx$, $v = e^x$. Apply formula: $\int x e^x\,dx = x e^x - \int e^x\,dx = x e^x - e^x + C$. Always check your answer by differentiating. The formula is also useful for cyclic integrals like $\int e^x\sin x\,dx$, where you apply it twice and solve for the integral. Remember to keep the constant of integration $C$ at the end.