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Question Number 120 by novrya last updated on 25/Jan/15
Evaluate ∫tan θ dθ
$${Evaluate}\:\int{tan}\:\theta\:{d}\theta \\ $$
Answered by ssahoo last updated on 06/Dec/14
∫tan θ dθ =∫((sin θ)/(cos θ))dθ substituting cos θ=y −sin θdθ=dy ∫((sin θ)/(cos θ))dθ=∫− (dy/y) =−ln ∣y∣+C =−ln ∣cos θ∣+C
$$\int\mathrm{tan}\:\theta\:\:{d}\theta \\ $$$$=\int\frac{\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta}{d}\theta \\ $$$$\mathrm{substituting}\:\mathrm{cos}\:\theta={y} \\ $$$$−\mathrm{sin}\:\theta{d}\theta={dy} \\ $$$$\int\frac{\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta}{d}\theta=\int−\:\frac{{dy}}{{y}} \\ $$$$=−\mathrm{ln}\:\mid{y}\mid+{C} \\ $$$$=−\mathrm{ln}\:\mid\mathrm{cos}\:\theta\mid+{C} \\ $$

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