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Question Number 58168 by maxmathsup by imad last updated on 19/Apr/19

$$find\int \frac{\sqrt{tanx}}{sin\left(2x\right)}dx$$

Answered by tanmay last updated on 19/Apr/19

$$\int \frac{\sqrt{tanx}}{2tanx}\times (1+{tan}^{2}x)dx$$$$\frac{1}{2}\int \frac{{sec}^{2}x}{\sqrt{tanx}}dx$$$$\frac{1}{2}\int {\left(tanx\right)}^{\frac{-1}{2}}d\left(tanx\right)$$$$\frac{1}{2}\times \frac{{\left(tanx\right)}^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+c$$$$\sqrt{tanx}+c$$

Commented by maxmathsup by imad last updated on 19/Apr/19

$$thankssir$$

Answered by malwaan last updated on 19/Apr/19

$$\int \frac{\sqrt{\tan x}}{2sinxcosx}dx$$$$=\int \frac{\sqrt{tanx}}{2\frac{sinx}{cosx}\times {cos}^{2}x}dx$$$$=\frac{1}{2}\int \frac{{sec}^{2}x\sqrt{tanx}}{tanx}dx$$$$=\frac{1}{2}\int \frac{{sec}^{2}x}{\sqrt{tanx}}dx=\frac{1}{2}\int {\left(tanx\right)}^{-\frac{1}{2}}{sec}^{2}xdx$$$$=\frac{1}{2}\frac{{\left(tanx\right)}^{\frac{1}{2}}}{\frac{1}{2}}+c=\sqrt{tanx}+c$$

Commented by maxmathsup by imad last updated on 19/Apr/19

$$thankssir$$

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