Introduction to Integration by Parts

By now we have a fairly thorough procedure for how to evaluate many basic integrals. However, although we can integrate [latex]\displaystyle\int x\sin\left({x}^{2}\right)dx[/latex] by using the substitution, [latex]u={x}^{2}[/latex], something as simple looking as [latex]\displaystyle\int x\sin{x}dx[/latex] defies us. Many students want to know whether there is a product rule for integration. There isn’t, but there is a technique based on the product rule for differentiation that allows us to exchange one integral for another. We call this technique integration by parts.