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Question Number 40251 by MJS last updated on 17/Jul/18

$$1.\int \frac{d\alpha }{\sin 2\alpha +\tan 3\alpha }=?$$$$2.\int \frac{d\beta }{\cos 2\beta +\cos 3\beta }=?$$$$3.\int \frac{d\gamma }{\sinh 2\gamma +\tanh 3\gamma }=?$$$$4.\int \frac{d\delta }{\cosh 2\delta +\cosh 3\delta }=?$$

Answered by maxmathsup by imad last updated on 26/Jul/18

$$4)letI=\int \frac{dx}{ch\left(2x\right)+ch\left(3x\right)}$$$$I=\int \frac{dx}{\frac{e^{2x}+e^{-2x}}{2}+\frac{e^{3x}+e^{-3x}}{2}}=\int \frac{2dx}{e^{2x}+e^{-2x}+e^{3x}+e^{-3x}}$$$${=}_{e^x=t}\int \frac{2}{t^2+t^{-2}+t^3+t^{-3}}\frac{dt}{t}=\int \frac{2dt}{t^3+t^{-1}+t^4+t^{-2}}$$$$=\int \frac{t^{2}dt}{t^2(t^3+t^{-1}+t^4+t^{-2})}=\int \frac{t^{2}dt}{t^5+t+t^6+1}=\int \frac{t^{2}dt}{t^6+t^5+t+1}$$$$rootsofp\left(t\right)=t^6+t^5+t+1$$$$t_1=-1\left(double\right)$$$$t_2\sim 0,809+0,587i\left(complex\right)$$$$t_3=0,809-0,587i\left(complex\right)$$$$t_4\sim -0,309+0,9511i\left(complex\right)$$$$t_5\sim -0,309-0,9511i\left(complex\right)$$$$p\left(t\right)={(t+1)}^2(t-t_2)(t-{\stackrel{-}{t}}_2)(t-t_4)(t-{\stackrel{-}{t}}_4)\Rightarrow$$$$F\left(t\right)=\frac{t^2}{{(t+1)}^2(t^2-2Re\left(t_2\right)t+\mid t_2{\mid }^2)(t^2-2Re\left(t_4\right)t+\mid t_4{\mid }^2)}$$$$=\frac{a}{t+1}+\frac{b}{{(t+1)}^2}+\frac{ct+d}{t^2-2Re\left(t_2\right)t+\mid t{\mid }^2}+\frac{et+f}{t^2-2Re\left(t_4\right)t+\mid t_4{\mid }^2}...becontinued...$$

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