Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integral. \(\int \frac{\sin x}{\sqrt{\cos ^{3} x}} d x\)

Step-by-Step Solution

Answer

The integral \(\int \frac{\sin x}{\sqrt{\cos ^{3} x}} d x\) is equal to \(-\frac{2\sqrt{\cos x}}{3} + C\).

Step 1: Substitution

Let \( u = \cos x \), then \( du/dx = -\sin x \). We can write \( dx = \frac{du}{-\sin x} \). Substituting into the integral will give \(\int \frac{\sin x}{\sqrt{u^{3}}} \frac{du}{-\sin x} \). This simplifies to \(-\int \frac{1}{\sqrt{u^{3}}}\, du\)

Step 2: Simplifying the integral

The integral \(-\int \frac{1}{\sqrt{u^{3}}}\, du\) can be rewritten as \(-\int u^{-3/2}\, du\). This is a simple power rule integral where the power is \(-3/2\). We add 1 to the power and divide by the new power.

Step 3: Apply the power rule

By applying the power rule, the integral becomes \(-\frac{2}{3}u^{-1/2} + C = -\frac{2\sqrt{u}}{3} + C\).

Step 4: Substituting \(u\) back

Substituting \(u = \cos x\) back, the integral becomes \(-\frac{2\sqrt{\cos x}}{3} + C\)

Key Concepts

Trigonometric Integration

Trigonometric integration involves integrating functions that contain trigonometric functions like sine, cosine, tangent, and their inverses. This type of integration often requires the use of trigonometric identities to simplify the integrand or convert it into a form that is more easily integrable.

For example, in the given exercise, the integral \(\int \frac{\sin x}{\sqrt{\cos^{3}x}}dx\) initially appears complex due to the presence of both sine and cosine functions, as well as the cube and square root. By using substitution, we can transform the trigonometric integral into a simpler non-trigonometric form that is easier to handle.

Substitution Method

The substitution method is a technique used to evaluate integrals by changing the variable of integration to simplify the integral. This method is particularly useful when the integrand is a composite function or when it involves complicated expressions that can be simplified through substitution.

In the provided step-by-step solution, a suitable substitution \(u = \cos x\) is made. The differential \(dx\) is expressed in terms of \(du\), which, when substituted into the integral, cancels out the \(\sin x\) in the numerator and denominator, leaving behind an integrand involving only \(u\) that is simpler to integrate. Care must be taken to also substitute the limits of integration if the integral is definite. The substitution method transforms our problem into a format that allows us to directly apply the power rule for integration.

Power Rule Integration

The power rule integration is a fundamental technique in calculus, used to evaluate the integral of power functions of the form \(x^n\). The rule states that, provided \(n eq -1\), the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), plus the constant of integration \(C\).

Applying this to our substitution-integrated function \(u^{-3/2}\), we add 1 to the exponent to get \(u^{-1/2}\), and divide by the new exponent, resulting in \(\frac{2}{3}u^{-1/2}\) as the solution to the integral. Last, we include a negative sign from the initial substitution and add the integration constant \(C\) to complete the integration process. It's worth noting that care must be taken with the constant of integration, particularly when dealing with definite integrals.