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