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I feel like my approach thus far is valid, but I've reached a spot where it feel like things are going wrong. I've changed the $1/\sin^2 \to \csc^2$ and distributed it to make it easier to integrate. $\sin^2 \times \csc^2 = 1$, so those cancel... but that leaves me with a constant $6$, which if my variable were $x$ I'd feel right at home. But this is $\theta$, and making that $6$ into a $6 \theta$ just seems like I've gone wrong somewhere.

Can anyone give me a push in the right direction?

Here's my approach so far:

$$ \eqalign { \text {Question: } & \int \frac { 7 - 6 \sin ^ 2 ( \theta ) } { \sin ^ 2 ( \theta ) } \, d \theta \\ \text {My answer: } & \int ( 7 - 6 \sin ^ 2 \theta ) ( \csc ^ 2 \theta ) \, d \theta \\ & \int ( 7 \csc ^ 2 \theta - 6 \sin ^ 2 \theta \csc ^ 2 \theta ) \, d \theta \\ & \int ( 7 \csc ^ 2 \theta - 6 ) \, d \theta \\ & 7 \cot \theta \> \cdots \hfil } $$

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    $\begingroup$ Strictly speaking, you're not changing the $6$ into a $6\theta$; you're changing the $6\,\mathrm d\theta$ into a $6\theta$. Maybe that will make you feel better about it. Anyway, the $6\theta$ is correct; subtract that from the $-7\cot\theta$ (and tack on a ${}+C$ if you're expected to have that in your answers), and you'll be done. $\endgroup$
    – Toby Bartels
    Aug 3, 2021 at 3:15
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    $\begingroup$ Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. $\endgroup$
    – José Carlos Santos
    Aug 3, 2021 at 3:16
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    $\begingroup$ I've submitted an edit with math formatting, so as soon as it gets approved by someone with enough reputation, you can open your question for editing and look at how I formatted it. See the link in José's comment above for more advice on how to do this in your questions. $\endgroup$
    – Toby Bartels
    Aug 3, 2021 at 3:35

1 Answer 1

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Yes, that $6$ becomes $6\theta$. That is$$\int7\csc^2(\theta)-6\,\mathrm d\theta=-7\cot(\theta)-6\theta+C.$$

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    $\begingroup$ There's a sign error here; it's $-6$, not $+6$. $\endgroup$
    – Toby Bartels
    Aug 3, 2021 at 3:17
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    $\begingroup$ @TobyBartels I've edited my answer. Thank you. $\endgroup$
    – José Carlos Santos
    Aug 3, 2021 at 3:19

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