Determine the integral of negative seven times the sin of four 𝑥 with respect to 𝑥.
We’re asked to evaluate the integral of a trigonometric function. And we know a lot of rules for helping us evaluate integrals of trigonometric functions. And to help us evaluate this, we need to recall the following standard trigonometric integral result. For any real constant 𝑎 not equal to zero, the integral of the sin of 𝑎𝑥 with respect to 𝑥 is equal to negative the cos of 𝑎𝑥 all divided by 𝑎 plus our constant of integration 𝐶.
We can see that our integral was almost in this form already. In fact, we can write this in this form directly. We just need to take the constant factor of negative seven outside of our integral. This means we’ve now rewritten the integral given to us in the question as negative seven times the integral of the sin of four 𝑥 with respect to 𝑥. And we can see this integral isn’t exactly in the form we need. We set our value of 𝑎 equal to four.
By setting our value of 𝑎 equal to four and remembering we’re multiplying our entire integral by negative seven, we get that this evaluates to give us negative seven times negative the cos of four 𝑥 over four plus 𝐶. The next thing we’ll do is distribute negative seven over our parentheses. Our first term will be negative seven times negative the cos of four 𝑥 divided by four. This simplifies to give us seven over four times the cos of four 𝑥. And then our second term will be negative seven times 𝐶.
We could leave our answer like this; however, it’s worth pointing out that 𝐶 is a constant of integration. This means when we originally called our variable 𝐶, we could’ve instead called it negative 𝐶 divided by seven. Then, when we multiplied through by negative seven, we would’ve instead got negative seven multiplied by negative 𝐶 divided by seven. And we can then simplify this to give us positive 𝐶. Sometimes you’ll see the constant of integration written as 𝑎 and then eventually rewritten as 𝐶. And sometimes you’ll see this step skipped altogether. It doesn’t matter which you prefer; it’s all personal preference.
Therefore, we were able to determine the integral of negative seven times the sin of four 𝑥 with respect to 𝑥 is equal to seven over four times the cos of four 𝑥 plus our constant of integration 𝐶.