Question

\[\int {\dfrac{1}{{\sin x\sqrt {\sin x\cos x} }}} dx = \]
A) \[\sqrt {\tan x} + c\]
B) \[2\sqrt {\tan x} + c\]
C) \[ - 2\sqrt {\cot x} + c\]
D) \[2\sqrt {\cot x} + c\]

Answer 1

Hint: The given function is indefinite since there is no limit given. Indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals.

Complete step by step answer:
 Let the given integral be $I$ such that:
\[I = \int {\dfrac{1}{{\sin x\sqrt {\sin x\cos x} }}} dx\]
Multiply numerator and the denominator of the function with \[{\sec ^2}x\], hence we get
\[I = \int {\dfrac{{{{\sec }^2}x}}{{\sin x\sqrt {\sin x\cos x} \times {{\sec }^2}x}}} dx\]
Now this can be written as
\[I = \int {\dfrac{{{{\sec }^2}x}}{{\tan x\sqrt {\tan x} }}} dx\]
Which is equivalent to
\[I = \int {\dfrac{{{{\sec }^2}x}}{{{{\tan }^{\dfrac{3}{2}}}x}}} dx\]
Now let
\[
  \sqrt {\tan x} = t \\
  \tan x = {t^2} \\
 \]
So by differentiating we get
\[
\Rightarrow \tan x = {t^2} \\
\Rightarrow {\sec ^2}x = 2t\dfrac{{dt}}{{dx}} \\
\Rightarrow dx = \dfrac{{2t}}{{{{\sec }^2}x}}dt \\
 \]
Now substitute the value of \[{\sec ^2}xdx = 2tdt\]in the integral function, hence we get
\[
  I = \int {\dfrac{{{{\sec }^2}x}}{{{t^3}}} \times \dfrac{{2tdt}}{{{{\sec }^2}x}}} \\
   = \int {\dfrac{2}{{{t^2}}}dt} \\
 \]
By integrating, we get
\[
  I = \int {\dfrac{2}{{{t^2}}}dt} \\
   = - \dfrac{2}{t} + c \\
 \]
Now substitute the value of\[t = \sqrt {\tan x} \], we get
\[
  I = - \dfrac{2}{t} + c \\
   = - \dfrac{2}{{\sqrt {\tan x} }} + c \\
 \]
This can be written as
\[
  I = - \dfrac{2}{{\sqrt {\tan x} }} + c \\
   = - 2\sqrt {\cot x} + c \\
 \]
Hence
$\Rightarrow$ \[\int {\dfrac{1}{{\sin x\sqrt {\sin x\cos x} }}} dx = - 2\sqrt {\cot x} + c\]. Hence Option 3 is correct.

Note: While substituting the real parameter of the question with the auxiliary parameter, one should be sure that it will not make the problem more complex. However, selecting an auxiliary parameter completely depends on the individual point of view.