# Video: MATH-DYNA-2018-S1-Q05

The velocity of a body is given by 𝑉 = 1 + sin (𝑡), and its displacement, 𝑥, is equal to −3 at 𝑡 = 0. Find 𝑥 as a function of time 𝑡.

02:21

### Video Transcript

The velocity of a body is given by 𝑉 is equal to one plus sin of 𝑡, and its displacement, 𝑥, is equal to negative three at 𝑡 is equal to zero. Find 𝑥 as a function of time 𝑡.

Now, we know that our velocity, 𝑉, is the change in displacement over time. Or we can say that 𝑉 is equal to d𝑥 by d𝑡 since d𝑥 by d𝑡 represents change in displacement over time. And since velocity is the differential of displacement with respect to time, this means that displacement must be equal to the integral of velocity with respect to time. So 𝑥 is equal to the integral of 𝑉 d𝑡. Now, we’re given in the question that 𝑉 is equal to one plus sin of 𝑡. So we can substitute this in for 𝑉 here. And this gives us that 𝑥 is equal to the integral of one plus sin of 𝑡 with respect to 𝑡.

The next step is to complete the integration. When we integrate one with respect to 𝑡, we simply get 𝑡. And then, when we integrate sin of 𝑡 with respect to 𝑡, we get negative cos of 𝑡. And since we have integrated here, we mustn’t forget to add our constant of integration, which we’ll call 𝐶. Now, we have found an equation for displacement in terms of time. We have 𝑥 is equal to 𝑡 minus cos of 𝑡 plus 𝐶. However, we do not know what 𝐶 is. We’re given in the question that the displacement, 𝑥, is equal to negative three at 𝑡 is equal to zero.

And so we can substitute in 𝑥 is equal to negative three and 𝑡 equals zero into our equation, giving us negative three is equal to zero minus cos of zero plus 𝐶. And now we know that cos of zero is simply one. This gives us that negative three is equal to negative one plus 𝐶. And we simply add one to both sides of the equation to find that 𝐶 is equal to negative two. So we have found our constant of integration, 𝐶, which is equal to negative two. And we can substitute it back into our equation for 𝑥 in terms of 𝑡. And we find that our displacement, 𝑥, as a function of time, 𝑡, is 𝑥 is equal to 𝑡 minus cos of 𝑡 minus two.