"How do I integrate sin⁡θ+sin⁡θtan2θ?" was answered by our experts

Connor Randall

Answered question

2022-01-27

How do I integrate

\sin θ + \sin θ \tan^{2} θ

Ydaxq

Beginner2022-01-28Added 12 answers

\sin θ + \sin θ \tan^{2} θ = \sin θ (1 + \tan^{2} θ) = \sin θ \sec^{2} θ = \sec θ \tan θ

because

(1 + \tan^{2} θ) = \sec^{2} θ

is a standard identity.
Also

\frac{d \sec θ}{d θ} = \sec θ \tan θ

is a standard derivative.
Therefore

\int (\sin θ + \sin θ \tan^{2} θ), d θ = \int \sec θ \tan θ, d θ = \sec θ + C

where C is an arbitrary constant

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