Question Video: Integrating Trigonometric Functions Mathematics

Video Transcript

Determine the integral of two times the sin of 𝑥 divided by six with respect to 𝑥.

We can see the question is asking us to determine the integral of a trigonometric function. And this is the form of a standard trigonometric integral rule which we should commit to memory. For constants 𝑎 and 𝑏, where 𝑏 is not equal to zero, the integral of 𝑎 times the sin of 𝑏𝑥 with respect to 𝑥 is equal to negative 𝑎 times the cos of 𝑏𝑥 divided by 𝑏 plus our constant of integration 𝑐.

First, in our integral, we can see we’re multiplying by two, so our value of 𝑎 is equal to two. Next, we can see we’re taking the sin of 𝑥 divided by six. We’ll write 𝑥 divided by six as one-sixth times 𝑥. And by writing it in this way, we can see that our value of 𝑏 is equal to one-sixth. So to integrate two times the sin of one over six times 𝑥 with respect to 𝑥, we just need to apply this integral rule. So by using our integral rule with 𝑎 equal to two and 𝑏 equal to one-sixth, we get negative two times the cos of one-sixth times 𝑥 divided by one-sixth plus a constant of integration 𝑐.

And we can simplify this answer. First, we’ll write one-sixth times 𝑥 as 𝑥 over six. Next, instead of dividing by one-sixth, we’ll multiply by the reciprocal of one-sixth. And the reciprocal of one-sixth is six. So this gives us negative two times six times the cos of 𝑥 over six plus 𝑐. And we can simplify negative two times six to give us negative 12. Therefore, we’ve shown the integral of two times the sin of 𝑥 over six with respect to 𝑥 is equal to negative 12 times the cos of 𝑥 over six plus 𝑐.