Menu Close

cos-1-5-x-sin-1-5-x-sinx-cosx-dx-cos-3-2-x-sin-1-2-x-cos-1-2-x-dx-sin-3-2-x-sin-1-2-x-cos-1-2-x-dx-cosx-sin-1-2-x-dx-sinx-cos-1-2-x-




Question Number 190452 by vishal1234 last updated on 03/Apr/23
∫((cos^(1.5) x−sin^(1.5) x)/( (√(sinx cosx)))) dx  = ∫((cos^(3/2) x)/(sin^(1/2) x cos^(1/2) x))dx−∫((sin^(3/2) x)/(sin^(1/2) x cos^(1/2) x)) dx  = ∫ ((cosx)/(sin^(1/2) x)) dx − ∫ ((sinx)/(cos^(1/2) x)) dx  = ∫ (dt/t^(1/2) ) − ∫ (((−dz))/z^(1/2) )  where cosx = z and sinx = t  = 2 (√(sinx)) + 2 (√(cosx)) + C
$$\int\frac{{cos}^{\mathrm{1}.\mathrm{5}} {x}−{sin}^{\mathrm{1}.\mathrm{5}} {x}}{\:\sqrt{{sinx}\:{cosx}}}\:{dx} \\ $$$$=\:\int\frac{{cos}^{\frac{\mathrm{3}}{\mathrm{2}}} {x}}{{sin}^{\mathrm{1}/\mathrm{2}} {x}\:{cos}^{\mathrm{1}/\mathrm{2}} {x}}{dx}−\int\frac{{sin}^{\frac{\mathrm{3}}{\mathrm{2}}} {x}}{{sin}^{\mathrm{1}/\mathrm{2}} {x}\:{cos}^{\mathrm{1}/\mathrm{2}} {x}}\:{dx} \\ $$$$=\:\int\:\frac{{cosx}}{{sin}^{\mathrm{1}/\mathrm{2}} {x}}\:{dx}\:−\:\int\:\frac{{sinx}}{{cos}^{\mathrm{1}/\mathrm{2}} {x}}\:{dx} \\ $$$$=\:\int\:\frac{{dt}}{{t}^{\mathrm{1}/\mathrm{2}} }\:−\:\int\:\frac{\left(−{dz}\right)}{{z}^{\mathrm{1}/\mathrm{2}} } \\ $$$${where}\:{cosx}\:=\:{z}\:{and}\:{sinx}\:=\:{t} \\ $$$$=\:\mathrm{2}\:\sqrt{{sinx}}\:+\:\mathrm{2}\:\sqrt{{cosx}}\:+\:{C} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *