Video Transcript
Determine the indefinite integral of seven cos of two 𝑥 over three with respect to 𝑥.
Our integrand is a trigonometric function. And whilst there is a general formula, we can quote to find the indefinite integral of cos of 𝑎𝑥 with respect to 𝑥. Let’s just look at where that comes from. We’re going to begin by recalling that the integral of cos of 𝑥 with respect to 𝑥 is sin 𝑥 plus some constant of integration 𝑐. And that’s simply because the antiderivative of cos of 𝑥 is sin 𝑥. When we differentiate sin 𝑥, we get cos of 𝑥. We’re going to begin by rewriting our integral slightly.
We recall that we can take any constant factors outside of the integral and focus on integrating the function in 𝑥 itself. So our integral is equal to seven times the indefinite integral of cos of two 𝑥 over three with respect to 𝑥. We’re looking to integrate cos of two 𝑥 over three. And we know that the integral of cos of 𝑥 with respect to 𝑥 is sin of 𝑥. So we’re going to perform a substitution. We’re going to let 𝑢 be equal to two 𝑥 over three. This means that d𝑢 by d𝑥 must be equal to two-thirds. And whilst d𝑢 by d𝑥 isn’t a fraction when we’re working with integration by substitution, we treat it a little like one. And we said that this is equivalent to three over two d𝑢 equals d𝑥.
We can now replace two 𝑥 over three with 𝑢 and d𝑥 with three over two d𝑢. And so, our integral is equal to seven times the integral of three over two cos of 𝑢 with respect to 𝑢. At this stage, we might choose to take the constant factor of three over two out. And we see it’s equal to 21 over two times that indefinite integral of cos of 𝑢 with respect to 𝑢. We know that the integral of cos of 𝑥 is sin 𝑥. So the integral of cos of 𝑢 must be sin 𝑢. We mustn’t forget to include that constant of integration 𝑐.
Distributing our parentheses, this becomes 21 over two sin 𝑢 plus a different constant of integration, capital 𝐶. We aren’t quite finished though. We were looking to integrate our function with respect to 𝑥. So we’re going to replace 𝑢 with that earlier substitution, with two 𝑥 over three. And when we do, we see that the indefinite integral of seven cos of two 𝑥 over three d𝑥 is 21 over two sin of two 𝑥 over three plus 𝐶.
Now, in fact, the general rule for integrating cos of 𝑎𝑥 with respect to 𝑥 is we get one over 𝑎 times sin of 𝑎𝑥 plus 𝐶. It is absolutely fine to quote this result. But it’s really useful to know where it comes from.
