Question

What is the integral of $\sqrt {\sin x} \cos x dx?$

Answer 1

In this integration question, the key observation is to substitute the sinx term to some variable and differentiate on both sides and then substitute the value of sinx and the equation after differentiation in the given integration to simplify it and then solve the integration with the help of known standard integration results.

Complete step-by-step solution:
The given integration is,
$\int {\sqrt {\sin x} \cos xdx}$
Let I= $\int {\sqrt {\sin x} \cos xdx}$
Let $\sin x = t$
On differentiating both sides with respect to x
$\Rightarrow \dfrac{{d(\sin x)}}{{dx}} = \dfrac{{d(t)}}{{dx}}$
As $\dfrac{{d(\sin x)}}{{dx}} = \cos x$
$\therefore \;{\text{cosx = }}\dfrac{{dt}}{{dx}}$
$\Rightarrow \cos xdx = dt$
Or,
$dt = \cos xdx$
On substituting values of sinx and cosxdx in I,
$\therefore {\text{ I = }}\int {\sqrt t dt}$
It can also be written as,
${\text{I = }}\int {{{\text{t}}^{\dfrac{1}{2}}}dt}$
As the equation is in the standard form of integration given by $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} + c$
Where c is integration constant.
On comparing,
$n = \dfrac{1}{2}$
$\therefore {\text{ I = }}\dfrac{{{{\text{t}}^{\dfrac{1}{2} + 1}}}}{{\dfrac{1}{2} + 1}} + C$
Where C is constant of integration.
On simplifying,
$I = \dfrac{{2{t^{\dfrac{3}{2}}}}}{3} + C$
On substituting the value of t,
$\Rightarrow I = \dfrac{{2{{(\sin x)}^{\dfrac{3}{2}}}}}{3} + C$

Note: This is trigonometry-based question and in order to solve this the identities related to it must be known. All the multiple and compound angle formulas should be known. The equation should be solved in accordance with the identities which would result in the correct solution. Calculations should be done carefully to avoid any mistake. Always try to solve the question step by step so that the wrong step can be determined and changed. The value of the constant can be found if the initial condition is given i.e. when x=0, the result of the integration would be some constant value, and by equating the value of the constant of integration can be found. This is indefinite-integral hence the constant of integration must be there but if it is definite-integral then the final answer would be some constant value. Also the results of the standard indefinite integral must be known so that it can be used directly.