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Question

Mathematics Question on integral

Integrate the function: tan⁡xsin⁡xcos⁡x\frac{\sqrt {\tan x}}{\sin x \cos x}

Answer

Let I=∫tan⁡xsin⁡xcos⁡xdxI= \int \frac{\sqrt {\tan x}}{\sin x \cos x}dx

= ∫tan⁡x∗cos⁡xsin⁡xcos⁡x∗cosxdx\int \frac{\sqrt {\tan x * \cos} x}{\sin x \cos x * cos x}dx

=∫tan⁡xtan⁡xcos⁡2xdx\int \frac{\sqrt {\tan x}}{\tan x \cos^2 x}dx

= ∫sec⁡2xdxtan⁡x\int \frac{\sec^2 xdx}{\sqrt {\tan x}}

Let tan x = t ⇒ sec2 xdx = dt

∴ I=∫dttI= \int \frac{dt}{\sqrt t}

= 2t+C2\sqrt t+C

= 2tan⁡x+C2 \sqrt {\tan x}+C