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

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

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

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

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

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

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