Video: Integrating Trigonometric Functions

Determine ∫ 3 cos 6𝑥 d𝑥.

01:47

Video Transcript

Determine the integral of three cos six 𝑥.

So the first thing we can do we’re gonna integrate this expression. Let’s take our constant, which is three, outside of the integration sign because this isn’t gonna affect our integration. And next, to enable us to integrate this expression, what we’re gonna use is the substitution method. So we’re gonna substitute 𝑢 is equal to six 𝑥.

So first of all, before we can do that, what we need to do is work out what d𝑥 is in terms of d𝑢. And to do that, we’re gonna differentiate 𝑢 with respect to 𝑥. So if we differentiate six 𝑥, we’re gonna get six. So we can say that d𝑢 d𝑥 is equal to six. So therefore, d𝑥 is equal to one over six d𝑢. So then, what we’ve got is three multiplied by the integral of a sixth cos 𝑢.

So once again, what we can do at this stage, we can take out our constant, which is a sixth, because again it’s not gonna affect our integral. So therefore, we’re gonna get a half multiplied by the integral of cos 𝑢. And we got half because we had three multiplied by a sixth, which is three-sixths, which is a half.

So now, if we integrate cos 𝑢, it’s gonna be straightforward cause we know that this is one of our standard integrals because the integral of cos 𝑥 is equal to sin 𝑥. So therefore, we’re gonna get a half sin 𝑢 plus 𝑐, where 𝑐 is our constant of integration. So now, what we need to do is substitute back in 𝑢 is equal to six 𝑥.

So therefore, when we do that, we can say that the integral of three cos six 𝑥 is equal to a half sin six 𝑥 plus 𝑐.

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