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Determine the integral of three cos six 𝑥.
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So the first thing we can do we’re gonna integrate this expression.
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Let’s take our constant, which is three, outside of the integration sign because this isn’t gonna affect our integration.
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And next, to enable us to integrate this expression, what we’re gonna use is the substitution method.
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So we’re gonna substitute 𝑢 is equal to six 𝑥.
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So first of all, before we can do that, what we need to do is work out what d𝑥 is in terms of d𝑢.
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And to do that, we’re gonna differentiate 𝑢 with respect to 𝑥.
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So if we differentiate six 𝑥, we’re gonna get six.
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So we can say that d𝑢 d𝑥 is equal to six.
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So therefore, d𝑥 is equal to one over six d𝑢.
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So then, what we’ve got is three multiplied by the integral of a sixth cos 𝑢.
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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.
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So therefore, we’re gonna get a half multiplied by the integral of cos 𝑢.
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And we got half because we had three multiplied by a sixth, which is three-sixths, which is a half.
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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.
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So therefore, we’re gonna get a half sin 𝑢 plus 𝑐, where 𝑐 is our constant of integration.
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So now, what we need to do is substitute back in 𝑢 is equal to six 𝑥.
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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.