# How do I integrate \sin\theta + \sin\theta \tan^2\theta?

How do I integrate $\mathrm{sin}\theta +\mathrm{sin}\theta {\mathrm{tan}}^{2}\theta$?
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Ydaxq
$\mathrm{sin}\theta +\mathrm{sin}\theta {\mathrm{tan}}^{2}\theta =\mathrm{sin}\theta \left(1+{\mathrm{tan}}^{2}\theta \right)=\mathrm{sin}\theta {\mathrm{sec}}^{2}\theta =\mathrm{sec}\theta \mathrm{tan}\theta$
because $\left(1+{\mathrm{tan}}^{2}\theta \right)={\mathrm{sec}}^{2}\theta$ is a standard identity.
Also
$\frac{d\mathrm{sec}\theta }{d\theta }=\mathrm{sec}\theta \mathrm{tan}\theta$
is a standard derivative.
Therefore
$\int \left(\mathrm{sin}\theta +\mathrm{sin}\theta {\mathrm{tan}}^{2}\theta \right),d\theta =\int \mathrm{sec}\theta \mathrm{tan}\theta ,d\theta =\mathrm{sec}\theta +C$
where C is an arbitrary constant