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Section 5.2 : Computing Indefinite Integrals

16. Evaluate \( \displaystyle \int{{12 + \csc \left( \theta \right)\left[ {\sin \left( \theta \right) + \csc \left( \theta \right)} \right]\,d\theta }}\).

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Hint : From previous problems in this set we should know how to deal with the product in the integrand.
Start Solution

Before doing the integral we need to multiply out the product and don’t forget the definition of cosecant in terms of sine.

\[\begin{align*}\int{{12 + \csc \left( \theta \right)\left[ {\sin \left( \theta \right) + \csc \left( \theta \right)} \right]\,d\theta }} & = \int{{12 + \csc \left( \theta \right)\sin \left( \theta \right) + {{\csc }^2}\left( \theta \right)\,d\theta }}\\ & = \int{{13 + {{\csc }^2}\left( \theta \right)\,d\theta }}\end{align*}\]

Recall that,

\[\csc \left( \theta \right) = \frac{1}{{\sin \left( \theta \right)}}\]

and so,

\[\csc \left( \theta \right)\sin \left( \theta \right) = 1\]

Doing this allows us to greatly simplify the integrand and, in fact, allows us to actually do the integral. Without this simplification we would not have been able to integrate the second term with the knowledge that we currently have.

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At this point there really isn’t too much to do other than to evaluate the integral.

\[\int{{12 + \csc \left( \theta \right)\left[ {\sin \left( \theta \right) + \csc \left( \theta \right)} \right]\,d\theta }} = \int{{13 + {{\csc }^2}\left( \theta \right)\,d\theta }} = \require{bbox} \bbox[2pt,border:1px solid black]{{13\theta - \cot \left( \theta \right) + c}}\]

Don’t forget that with trig functions some terms can be greatly simplified just by recalling the definition of the trig functions and/or their relationship with the other trig functions.