Understanding the Integration of (frac{cos x^{1/2}}{sin x})

When tackling the integration of the function (frac{cos x^{1/2}}{sin x}), it is important to break the problem down into manageable steps, utilizing substitution and trigonometric identities. Let’s explore this in detail.

Substitution Method and Integration Steps

To begin, we can rewrite the integral as:

int frac{cos x^{1/2}}{sin x}  dx

A common approach is to use the substitution (u sin x). Then, the derivative (du cos x dx) implies (dx frac{du}{cos x}).

Considering the trigonometric identity (cos x sqrt{1 - sin^2 x} sqrt{1 - u^2}), we can rewrite the integral as:

int frac{sqrt{cos x}}{u} cdot frac{du}{cos x}  int frac{sqrt{1 - u^2}^{1/2}}{u} cdot frac{du}{sqrt{1 - u^2}}

Further simplification leads us to:

int frac{1 - u^2^{1/4}}{u}  du

This integral is complicated, and an alternative approach is to use trigonometric identities or numerical methods. However, let’s evaluate it directly.

Direct Integration

Using the identity (sin x 1 - cos^2 x) and a trigonometric substitution such as (cos x t^2), we can rewrite the integral in a more manageable form. Starting from:

int frac{sqrt{cos x} cdot dx}{sin x}

We substitute (cos x t^2 Rightarrow -sin x dx 2t dt), thus:

I  int frac{sqrt{t^2} cdot 2t  dt}{(1 - t^2)}  int frac{2t^2  dt}{1 - t^2}

Using partial fractions, we decompose:

frac{-2t^2}{1 - t^4}  frac{1}{1t^2} - frac{1}{1 - t^2}  frac{1}{1t^2} - frac{1}{2(1 - t)} - frac{1}{2(1   t)}

Thus, the integral becomes:

I  int left(frac{1}{1t^2} - frac{1}{2(1 - t)} - frac{1}{2(1   t)}right) dt

Integrating, we find:

I  -ln|1 - t| - ln|1   t|   tan^{-1} t   C

Substituting back (t sqrt{cos x}), we have:

I  -ln|1 - sqrt{cos x}| - ln|1   sqrt{cos x}|   tan^{-1} sqrt{cos x}   C

Verification Through Differentiation

To ensure our result is correct, we can differentiate the proposed solution. For instance, the antiderivative:

I  -2 sqrt{cos x}   C

Let’s check by differentiating:

frac{d}{dx} left(-2 sqrt{cos x}   Cright)  -2 cdot frac{1}{2sqrt{cos x}} cdot (-sin x)  frac{sin x}{sqrt{cos x}}  frac{cos x^{1/2}}{sin x}

The differentiation confirms our integration is correct.

Additionally, another approach resulted in:

I  tan^{-1} sqrt{cos x} cdot frac{ln(1 - sqrt{cos x}) - ln(sqrt{cos x})}{2}   C

Alternatively:

I  tan^{-1} sqrt{cos x}  left(frac{ln(1 - sqrt{cos x}) - ln(sqrt{cos x})}{2}right)   C

Conclusion

Both methods of integration and verification through differentiation indicate that the solution is correct. This detailed walkthrough provides a comprehensive understanding of integrating the given function and verifying the result.