Question Number 68869 by mathmax by abdo last updated on 16/Sep/19
![let f(a) =∫_0 ^(π/2) (dx/(a+sinx)) (a real) 1)find a explicit form for f(a) 2) calculste also g(a)=∫_0 ^(π/2) (dx/((a+sinx)^2 )) and h(a)=∫_0 ^(π/2) (dx/((a+sinx)^3 )) 3)give f^((n)) (a) at form of integral 4) find the values of integrals ∫_0 ^(π/2) (dx/(3+sinx)) , ∫_0 ^(π/2) (dx/((3+sinx)^2 )) and ∫_0 ^(π/2) (dx/((3+sinx)^3 ))](https://www.tinkutara.com/question/Q68869.png)
$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{{a}+{sinx}}\:\:\:\:\:\left({a}\:{real}\right) \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{explicit}\:{form}\:\:{for}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculste}\:{also}\:{g}\left({a}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\left({a}+{sinx}\right)^{\mathrm{2}} }\:\:{and}\:{h}\left({a}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{dx}}{\left({a}+{sinx}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({a}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\mathrm{3}+{sinx}}\:,\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\left(\mathrm{3}+{sinx}\right)^{\mathrm{2}} } \\ $$$${and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\left(\mathrm{3}+{sinx}\right)^{\mathrm{3}} } \\ $$