What is the integral of (12+csc(θ)(sin(θ)+csc(θ))) ?

The integral of (12+csc(θ)(sin(θ)+csc(θ))) is -cot(θ)+13θ+C

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integral of (12+csc(θ)(sin(θ)+csc(θ)))

\int (12 + csc (θ) (sin (θ) + csc (θ))) d θ

- cot (θ) + 13 θ + C

Solution steps

\int 12 + csc (θ) (sin (θ) + csc (θ)) d θ

Expand

csc (θ) (sin (θ) + csc (θ)) : csc (θ) sin (θ) + csc^{2} (θ)

= \int 12 + csc (θ) sin (θ) + csc^{2} (θ) d θ

Simplify

12 + csc (θ) sin (θ) + csc^{2} (θ) : csc^{2} (θ) + 13

= \int csc^{2} (θ) + 13 d θ

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int csc^{2} (θ) d θ + \int 13 d θ

\int csc^{2} (θ) d θ = - cot (θ)

\int 13 d θ = 13 θ

= - cot (θ) + 13 θ

Add a constant to the solution

= - cot (θ) + 13 θ + C

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What is the integral of (12+csc(θ)(sin(θ)+csc(θ))) ?
The integral of (12+csc(θ)(sin(θ)+csc(θ))) is -cot(θ)+13θ+C