Understanding the Trigonometric Integral Problem

We are asked to evaluate the indefinite integral: \(\int (\sin x)^{-1/2} (\cos x)^{-3/2}dx\).

This integral involves trigonometric functions raised to fractional and negative powers. To solve this, we need to manipulate the expression to a form that is easier to integrate, often by using trigonometric identities or substitution methods.

Rewriting the Integrand

First, let's rewrite the integrand using positive exponents and radical notation:

The integrand is \((\sin x)^{-1/2} (\cos x)^{-3/2} = \frac{1}{(\sin x)^{1/2} (\cos x)^{3/2}} = \frac{1}{\sqrt{\sin x} \cdot (\cos x) \sqrt{\cos x}}\).

The expression is \(\frac{1}{\sqrt{\sin x \cos x} \cdot \cos x}\). This form doesn't immediately suggest a standard integration technique.

Let's try rewriting the entire denominator in terms of powers of \(\cos x\):

\(\frac{1}{(\sin x)^{1/2} (\cos x)^{3/2}}\)

We can try to transform this into a function of \(\tan x\) by dividing the numerator and denominator by a suitable power of \(\cos x\). The total power in the denominator is \(1/2 + 3/2 = 4/2 = 2\). Let's divide by \(\cos^2 x\).

Transforming the Integral using \(\cos^2 x\)

Divide the numerator and the denominator of the integrand by \(\cos^2 x\):

\(\frac{1}{(\sin x)^{1/2} (\cos x)^{3/2}} = \frac{1 / \cos^2 x}{(\sin x)^{1/2} (\cos x)^{3/2} / \cos^2 x}\)

The numerator becomes \(\sec^2 x\).

The denominator becomes: \((\sin x)^{1/2} (\cos x)^{3/2} (\cos x)^{-2} = (\sin x)^{1/2} (\cos x)^{3/2 - 2} = (\sin x)^{1/2} (\cos x)^{-1/2}\) \(= \frac{(\sin x)^{1/2}}{(\cos x)^{1/2}} = \left(\frac{\sin x}{\cos x}\right)^{1/2} = (\tan x)^{1/2}\)

So, the integral becomes:

\(\int \frac{\sec^2 x}{(\tan x)^{1/2}} dx = \int (\tan x)^{-1/2} \sec^2 x \, dx\)

Using Substitution for Integration

Now, the integral is in a form suitable for substitution. Let \(u = \tan x\). Then, the differential \(du\) is the derivative of \(u\) with respect to \(x\), multiplied by \(dx\): \(du = \frac{d}{dx}(\tan x) \, dx = \sec^2 x \, dx\)

Substitute \(u\) and \(du\) into the integral:

\(\int (\tan x)^{-1/2} \sec^2 x \, dx = \int u^{-1/2} du\)

Evaluating the Integral with Substitution

Now we integrate \(u^{-1/2}\) with respect to \(u\). We use the power rule for integration: \(\int u^n du = \frac{u^{n+1}}{n+1} + C\), where \(n \neq -1\).

Here, \(n = -1/2\). So, \(n+1 = -1/2 + 1 = 1/2\).

\(\int u^{-1/2} du = \frac{u^{-1/2+1}}{-1/2+1} + c = \frac{u^{1/2}}{1/2} + c = 2 u^{1/2} + c\)

Substituting Back to Original Variable

Finally, substitute back \(u = \tan x\) into the result:

\(2 u^{1/2} + c = 2 (\tan x)^{1/2} + c = 2 \sqrt{\tan x} + c\)

Conclusion

The integral \(\int (\sin x)^{-1/2} (\cos x)^{-3/2}dx\) is equal to \(2\sqrt {\tan x}+ c\).

Revision Table: Key Calculus Concepts

Additional Information on Trigonometric Integrals

Integrals involving powers of sine and cosine can often be solved using various strategies:

• If the powers are integers, identities like \(\sin^2 x + \cos^2 x = 1\), \(\sin^2 x = \frac{1 - \cos(2x)}{2}\), or \(\cos^2 x = \frac{1 + \cos(2x)}{2}\) are useful.
• If the integrand is a rational function of sine and cosine, a substitution like \(t = \tan(x/2)\) (Weierstrass substitution) can transform the integral into a rational function of \(t\).
• In cases like the one solved, where fractional powers are involved, manipulating the integrand to isolate a function and its derivative (like \(\tan x\) and \(\sec^2 x\)) is a common and effective technique. Looking for combinations that relate to derivatives of basic trigonometric functions (\(\tan x \implies \sec^2 x\), \(\cot x \implies -\csc^2 x\), \(\sec x \implies \sec x \tan x\), \(\csc x \implies -\csc x \cot x\)) is key.
• Sometimes, integration by parts might be required for more complex trigonometric integrals.

Recognizing the structure of the integrand and which substitution or identity is applicable is crucial for solving trigonometric integrals efficiently.