Determine the integral of negative four times the cos of nine 𝑥 with respect to 𝑥.
We’re asked to evaluate the integral of a trigonometric function. And we can do this by recalling our rules for integrating trigonometric functions. We recall for any real constants 𝑎 and 𝑘 where 𝑎 is not equal to zero, the integral of 𝑘 times the cos of 𝑎𝑥 with respect to 𝑥 is equal to 𝑘 times the sin of 𝑎𝑥 divided by 𝑎 plus the constant of integration 𝐶. Integrals of this form come up a lot, so it’s worth committing these to memory.
We want to apply this to the integral given to us in the question. We can see our value of the constant 𝑘 is negative four and the value of the constant 𝑎 is nine. So by setting 𝑘 equal to negative four and 𝑎 equal to nine into our integral result, we get that our integral is equal to negative four sin of nine 𝑥 all divided by nine. And remember, we need to add our constant of integration 𝐶. And we’ll rewrite this as negative four over nine times the sin of nine 𝑥 plus 𝐶.
Therefore, by using our standard trigonometric integral results, we were able to show the integral of negative four times the cos of nine 𝑥 with respect to 𝑥 is equal to negative four over nine times the sin of nine 𝑥 plus 𝐶.