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Question Number 216990 by MathematicalUser2357 last updated on 26/Feb/25

$$\left(1\right)\int ({\sec }^{2}x\cdot \sqrt{\tan x})dx=?$$

Answered by Frix last updated on 26/Feb/25

$$\int {\sec }^{2}x\sqrt{\tan x}dx\stackrel{[t=\sqrt{\tan x}]}{=}2\int t^{2}dt=\frac{2}{3}{t}^3=$$$$=\frac{2}{3}{\tan }^{\frac{3}{2}}x+C$$

Commented by MathematicalUser2357 last updated on 27/Feb/25

$$\mathrm{what}\mathrm{about}\mathrm{the}\mathrm{second}\mathrm{question}?$$

Answered by MATHEMATICSAM last updated on 27/Feb/25

$$\mathrm{Let}\tan x=t$$$$\Rightarrow {\sec }^{2}xdx=dt$$$$$$$$\int \left({\sec }^{2}x\sqrt{\tan x}\right)dx$$$$=\int \sqrt{t}dt$$$$=\frac{2}{3}{t}^{\frac{3}{2}}+C$$$$=\frac{2}{3}{\tan }^{\frac{3}{2}}x+C$$

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