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From what I've gathered, every text book says we can do trig substitution with either sin, sec or tan. Some other examples use tan(x/2), but so far I've never encountered cosine substitution. Is there some reason why it's omitted, or is it just interchangeable with sin? https://en.wikipedia.org/wiki/Trigonometric_substitution

$$x=\sin \theta: \sqrt{1-x^2}=\sqrt{1-\sin^2 \theta}=\sqrt{\cos^2 \theta}=\cos(\theta)$$

$$x=\cos \theta:\sqrt{1-x^2}=\sqrt{1-\cos^2 \theta}=\sqrt{\sin^2 \theta}=\sin(\theta)$$

From the example above, it looks like I could use either when integrating.

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    $\begingroup$ It is interchangeable with sine. $\endgroup$ Oct 24, 2021 at 1:25
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    $\begingroup$ As long as it is a strictly increasing/decreasing continuous function you're using for the substitution, then yes it works $\endgroup$
    – Ariana
    Oct 24, 2021 at 1:33
  • $\begingroup$ Write $x = \sin \theta$ with \theta instead, and $\cos \theta, \sin \theta$ as \cos \theta, \sin \theta. For more, you can see the MathJax reference. $\endgroup$
    – Toby Mak
    Oct 24, 2021 at 2:37

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Like you said, both $\cos \theta$ and $\sin \theta$ work. The domains are a little different: $\sin \theta$ is one-to-one when $t \in [-\pi/2, \pi/2]$ and so on, whereas $\cos \theta$ is one-to-one when $t \in [-\pi, 0]$ or $[0, \pi]$.

However, $x = \cos \theta$, $dx = -\sin \theta \ d \theta$ so you have an extra negative sign. If you forget the negative sign, you might spend ages poring over your work.

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