# Video: Finding the Integral of a Function Using Integration by Parts

Use integration by parts to evaluate ∫ 𝑥 sin 𝑥 d𝑥.

02:10

### Video Transcript

Use integration by parts to evaluate the integral of 𝑥 times sin 𝑥 with respect to 𝑥.

The function we’re looking to integrate is the product of two functions. That’s 𝑥 and sin 𝑥. This indicates that we might need to use integration by parts to evaluate our integral. Remember the formula here says the integral of 𝑢 times d𝑣 by d𝑥 is equal to 𝑢𝑣 minus the integral of 𝑣 times d𝑢 by d𝑥. If we compare this formula to our integrand, we see that we’re going to need to decide which function is 𝑢. And which function is d𝑣 by d𝑥. So how do we decide this? Well, our aim is going to be to ensure that the second integral we get over here is a little simpler. We therefore want 𝑢 to be a function that either becomes simpler when differentiated or helps to simplify the integrand when multiplied by the function 𝑣. It should be quite clear that, out of 𝑥 and sin 𝑥, the function that becomes simpler when differentiated is 𝑥. So we let 𝑢 be equal to 𝑥 and d𝑣 by d𝑥 be equal to sin 𝑥. The derivative of 𝑢 with respect to 𝑥 is one. And what do we do with d𝑣 by d𝑥? Well, we’re going to need to find 𝑣. So we find the antiderivative of sin of 𝑥, which is of course negative cos of 𝑥.

Let’s substitute what we have into our formula. We see that our integral is equal to 𝑥 times negative cos of 𝑥 minus the integral of negative cos of 𝑥 times one d𝑥. This simplifies to negative 𝑥 cos of 𝑥. And then we take the factor of negative one outside of our integral. And we see that we add the integral of cos of 𝑥 evaluated with respect to 𝑥. The antiderivative of cos of 𝑥 is sin of 𝑥. And of course, because this is an indefinite integral, we must add that constant of integration 𝑐. And we see that the solution is negative 𝑥 cos of 𝑥 plus sin of 𝑥 plus 𝑐. Now it’s useful to remember that we can check our answer by differentiating. If we do, we indeed obtain 𝑥 times sin 𝑥 as required.