Integration by parts is one of the fundamental techniques in calculus. The proof begins with the product rule for differentiation. When we have two functions u and v, their product's derivative follows the product rule. By integrating both sides and rearranging, we derive the integration by parts formula.
The foundation of integration by parts lies in the product rule of differentiation. When we differentiate the product of two functions u and v, we get u prime times v plus u times v prime. This fundamental rule from differential calculus will be the key to deriving our integration formula.
The next step is to integrate both sides of the product rule equation. When we integrate the left side, we get back to the original product uv by the fundamental theorem of calculus. On the right side, we can split the integral into two separate integrals. This gives us the equation uv equals the integral of u prime v dx plus the integral of uv prime dx.
Now we rearrange the equation to isolate the integral of uv prime dx. We subtract the integral of u prime v dx from both sides. This gives us the integral of uv prime dx equals uv minus the integral of u prime v dx. This is almost our final integration by parts formula.
Finally, we convert our result to the standard differential notation. We let du equal u prime dx and dv equal v prime dx. Substituting these into our equation gives us the integration by parts formula: the integral of u dv equals uv minus the integral of v du. This completes our proof and gives us one of the most important integration techniques in calculus.