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Question Number 63117 by mugan deni last updated on 29/Jun/19
∫((cos x)/(2+3sin x+sin^2 x))dx
$$\int\frac{\mathrm{cos}\:{x}}{\mathrm{2}+\mathrm{3sin}\:{x}+\mathrm{sin}\:^{\mathrm{2}} {x}}{dx} \\ $$
Commented by mathmax by abdo last updated on 30/Jun/19
let I =∫ ((cosx)/(sin^2 x+3sinx +2))dx changement sinx =t give cosxdx =dt ⇒ I =∫ (dt/(t^2 +3t +2)) let solve t^2 +3t+2 =0 →Δ=9−8=1 ⇒ t_1 =((−3+1)/2) =−1 and t_2 =((−3−1)/2) =−2 ⇒t^2 +3t +2 =(t+1)(t+2) ⇒ I =∫ (dt/((t+1)(t+2))) = ∫ ((1/(t+1))−(1/(t+2)))dt =ln∣t+1∣−ln∣t+2∣ +c =ln∣1+sinx∣−ln∣2+sinx∣ +c .
$${let}\:{I}\:=\int\:\:\frac{{cosx}}{{sin}^{\mathrm{2}} {x}+\mathrm{3}{sinx}\:+\mathrm{2}}{dx}\:\:{changement}\:{sinx}\:={t}\:{give}\:{cosxdx}\:={dt}\:\Rightarrow \\ $$$${I}\:=\int\:\:\:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+\mathrm{3}{t}\:+\mathrm{2}}\:\:{let}\:{solve}\:{t}^{\mathrm{2}} \:+\mathrm{3}{t}+\mathrm{2}\:=\mathrm{0}\:\rightarrow\Delta=\mathrm{9}−\mathrm{8}=\mathrm{1}\:\Rightarrow \\ $$$${t}_{\mathrm{1}} =\frac{−\mathrm{3}+\mathrm{1}}{\mathrm{2}}\:=−\mathrm{1}\:\:{and}\:{t}_{\mathrm{2}} =\frac{−\mathrm{3}−\mathrm{1}}{\mathrm{2}}\:=−\mathrm{2}\:\Rightarrow{t}^{\mathrm{2}} \:+\mathrm{3}{t}\:+\mathrm{2}\:=\left({t}+\mathrm{1}\right)\left({t}+\mathrm{2}\right)\:\Rightarrow \\ $$$${I}\:=\int\:\:\frac{{dt}}{\left({t}+\mathrm{1}\right)\left({t}+\mathrm{2}\right)}\:=\:\int\:\:\:\left(\frac{\mathrm{1}}{{t}+\mathrm{1}}−\frac{\mathrm{1}}{{t}+\mathrm{2}}\right){dt}\:={ln}\mid{t}+\mathrm{1}\mid−{ln}\mid{t}+\mathrm{2}\mid\:+{c} \\ $$$$={ln}\mid\mathrm{1}+{sinx}\mid−{ln}\mid\mathrm{2}+{sinx}\mid\:+{c}\:. \\ $$
Answered by Hope last updated on 29/Jun/19
a=sinx da=cosxdx ∫(da/(a^2 +3a+2)) ∫(da/((a+1)(a+2))) ∫(((a+2)−(a+1))/((a+1)(a+2)))da ∫(da/(a+1))−∫(da/(a+2)) ln(a+1)−ln(a+2)+c ln(((a+1)/(a+2)))+c ln(((1+sinx)/(2+sinx)))+c
$${a}={sinx}\:{da}={cosxdx} \\ $$$$\int\frac{{da}}{{a}^{\mathrm{2}} +\mathrm{3}{a}+\mathrm{2}} \\ $$$$\int\frac{{da}}{\left({a}+\mathrm{1}\right)\left({a}+\mathrm{2}\right)} \\ $$$$\int\frac{\left({a}+\mathrm{2}\right)−\left({a}+\mathrm{1}\right)}{\left({a}+\mathrm{1}\right)\left({a}+\mathrm{2}\right)}{da} \\ $$$$\int\frac{{da}}{{a}+\mathrm{1}}−\int\frac{{da}}{{a}+\mathrm{2}} \\ $$$${ln}\left({a}+\mathrm{1}\right)−{ln}\left({a}+\mathrm{2}\right)+{c} \\ $$$${ln}\left(\frac{{a}+\mathrm{1}}{{a}+\mathrm{2}}\right)+{c} \\ $$$${ln}\left(\frac{\mathrm{1}+{sinx}}{\mathrm{2}+{sinx}}\right)+{c} \\ $$$$ \\ $$$$ \\ $$

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