we want to integrate 2x sine x with respects to x this doesn't fit a basic integration formula and integration using substitution doesn't work either so now we'll apply integration by parts here's the formula for integration by parts we'll let part of the integral equal u and the other part equal dv once we choose u we'll differentiate to determine differential u and once we choose dv we'll integrate to determine v and the guidelines tell us to select a u so that differential u is simpler than u and to select a dv that is easy to integrate and for this example that may not be a lot of help because both of these are easy to integrate and differentiate but if we did choose u equals two x differential u would be simpler it would just be two dx and if we selected u equals sine x differential u would be cosine x dx which is not any simpler so that means that dv is going to have to be sine x dx so now we'll have to integrate to determine v well the integral of sine x with respects to x would be negative cosine x now that we have everything we can apply the integration by parts formula and hopefully the integral of v d u is going to be easier to integrate and i think it will be so the original integral is equal to u times v well here's u and here's v so we'll have negative two x cosine x minus the integral of v d u well here's v and here's d u so we'll have the integral of negative two cosine x dx so we'll factor the negative two out and we have cosine x dx so we'll have negative two x cosine x this will be plus two times the antiderivative of cosine x which is sine x and then plus c and we could leave it like this but let's go ahead and factor a 2 out if we factor out a positive 2 we'd probably put sine x first minus x cosine x and then plus c so this would be our antiderivative using integration by parts in the last example we'll take a look at using integration by parts twice to determine an antiderivative i hope this was helpful