∫((4dx)/(3−5sin x))


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Question Number 136466 by aurpeyz last updated on 22/Mar/21

$\int \frac{4 d x}{3 - 5 s i n x}$
	Answered by mathmax by abdo last updated on 22/Mar/21

	$Φ = \int \frac{4 dx}{3 - 5 sinx} we do the changement \tan (\frac{x}{2}) = t \Rightarrow$ $Φ = \int \frac{4}{(3 - 5 \frac{2 t}{1 + t^{2}})} \frac{2 dt}{(1 + t^{2})} = 8 \int \frac{dt}{3 + {3 t}^{2} - 10 t}$ $= 8 \int \frac{dt}{{3 t}^{2} - 10 t + 3}$ $Δ^{^{'}} = 25 - 9 = 16 \Rightarrow t_{1} = \frac{5 + 4}{3} = 3 and t_{2} = \frac{5 - 4}{3} = \frac{1}{3} \Rightarrow$ $Φ = 8 \int \frac{dt}{3 (t - 3) (t - \frac{1}{3})} = \frac{8}{3} \times \frac{3}{8} \int (\frac{1}{t - 3} - \frac{1}{t - \frac{1}{3}}) dt$ $= \ln ∣ \frac{t - 3}{t - \frac{1}{3}} ∣ + C = \ln ∣ \frac{3 (t - 3)}{3 t - 1} ∣ + C = \ln ∣ \frac{t - 3}{3 t - 1} ∣ + C$ $Φ = \ln ∣ \frac{\tan (\frac{x}{2}) - 3}{3 \tan (\frac{x}{2}) - 1} ∣ + C$
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