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Question Number 177194 by peter frank last updated on 02/Oct/22

∫ (dx/((sin x)^((14)/9) (cos x)^(4/9) ))

$$\int\:\:\frac{\mathrm{dx}}{\left(\mathrm{sin}\:\mathrm{x}\right)^{\frac{\mathrm{14}}{\mathrm{9}}} \left(\mathrm{cos}\:\mathrm{x}\right)^{\frac{\mathrm{4}}{\mathrm{9}}} } \\ $$

Answered by som(math1967) last updated on 02/Oct/22

∫((sec^2 xdx)/((sinx)^((14)/9) ×(cosx)^(4/9) ×sec^2 x)) ∫((sec^2 xdx)/((sinx)^((14)/9) ×(cosx)^((4/9)−2) )) ∫((sec^2 xdx)/((tanx)^((14)/9) )) ∫(tanx)^(−((14)/9)) d(tanx) =(1/(1−((14)/9)))(tanx)^(1−((14)/9)) +C =−(9/(5(tanx)^(5/9) )) +C

$$\int\frac{{sec}^{\mathrm{2}} {xdx}}{\left({sinx}\right)^{\frac{\mathrm{14}}{\mathrm{9}}} ×\left({cosx}\right)^{\frac{\mathrm{4}}{\mathrm{9}}} ×{sec}^{\mathrm{2}} {x}} \\ $$$$\int\frac{{sec}^{\mathrm{2}} {xdx}}{\left({sinx}\right)^{\frac{\mathrm{14}}{\mathrm{9}}} ×\left({cosx}\right)^{\frac{\mathrm{4}}{\mathrm{9}}−\mathrm{2}} } \\ $$$$\int\frac{{sec}^{\mathrm{2}} {xdx}}{\left({tanx}\right)^{\frac{\mathrm{14}}{\mathrm{9}}} } \\ $$$$\:\int\left({tanx}\right)^{−\frac{\mathrm{14}}{\mathrm{9}}} {d}\left({tanx}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{14}}{\mathrm{9}}}\left({tanx}\right)^{\mathrm{1}−\frac{\mathrm{14}}{\mathrm{9}}} \:+{C} \\ $$$$=−\frac{\mathrm{9}}{\mathrm{5}\left({tanx}\right)^{\frac{\mathrm{5}}{\mathrm{9}}} }\:+{C} \\ $$

Commented by peter frank last updated on 02/Oct/22

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

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