# Video: Integrating Trigonometric Functions

Determine β« 7 cos (2π₯/3) dπ₯.

02:22

### Video Transcript

Determine the indefinite integral of seven cos of two π₯ over three with respect to π₯.

Our integrand is a trigonometric function. And whilst there is a general formula, we can quote to find the indefinite integral of cos of ππ₯ with respect to π₯. Letβs just look at where that comes from. Weβre going to begin by recalling that the integral of cos of π₯ with respect to π₯ is sin π₯ plus some constant of integration π. And thatβs simply because the antiderivative of cos of π₯ is sin π₯. When we differentiate sin π₯, we get cos of π₯. Weβ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 π₯ itself. So our integral is equal to seven times the indefinite integral of cos of two π₯ over three with respect to π₯. Weβre looking to integrate cos of two π₯ over three. And we know that the integral of cos of π₯ with respect to π₯ is sin of π₯. So weβre going to perform a substitution. Weβre going to let π’ be equal to two π₯ over three. This means that dπ’ by dπ₯ must be equal to two-thirds. And whilst dπ’ by dπ₯ isnβt a fraction when weβre working with integration by substitution, we treat it a little like one. And we said that this is equivalent to three over two dπ’ equals dπ₯.

We can now replace two π₯ over three with π’ and dπ₯ with three over two dπ’. And so, our integral is equal to seven times the integral of three over two cos of π’ with respect to π’. At this stage, we might choose to take the constant factor of three over two out. And we see itβs equal to 21 over two times that indefinite integral of cos of π’ with respect to π’. We know that the integral of cos of π₯ is sin π₯. So the integral of cos of π’ must be sin π’. We mustnβt forget to include that constant of integration π.

Distributing our parentheses, this becomes 21 over two sin π’ plus a different constant of integration, capital πΆ. We arenβt quite finished though. We were looking to integrate our function with respect to π₯. So weβre going to replace π’ with that earlier substitution, with two π₯ over three. And when we do, we see that the indefinite integral of seven cos of two π₯ over three dπ₯ is 21 over two sin of two π₯ over three plus πΆ.

Now, in fact, the general rule for integrating cos of ππ₯ with respect to π₯ is we get one over π times sin of ππ₯ plus πΆ. It is absolutely fine to quote this result. But itβs really useful to know where it comes from.