Question

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible. \(\int_{\pi / 6}^{\pi / 3} \frac{\cos ^{3} x}{\sqrt{\sin x}} d x\) (Round this answer to three decimal places.)

Short answer

0.569

Step-by-step solution

Identify the Problem Type

We need to evaluate the definite integral \( \int_{\pi/6}^{\pi/3} \frac{\cos^3 x}{\sqrt{\sin x}} \, dx \). This involves integration techniques and possibly substitution to simplify the expression.

Use Trigonometric Identity

Recall that \( \cos^3 x = (\cos^2 x) \cos x = (1 - \sin^2 x) \cos x \) using the identity \( \cos^2 x = 1 - \sin^2 x \). Substitute this into the integral to get: \[ \int_{\pi/6}^{\pi/3} \frac{(1 - \sin^2 x) \cos x}{\sqrt{\sin x}} \, dx.\]

Apply Substitution

Let \( u = \sin x \), so \( du = \cos x \, dx \). The limits of integration change as follows: when \( x = \pi/6 \), \( u = \sin(\pi/6) = 1/2 \); when \( x = \pi/3 \), \( u = \sin(\pi/3) = \sqrt{3}/2 \). The integral becomes: \[ \int_{1/2}^{\sqrt{3}/2} \frac{1 - u^2}{\sqrt{u}} \, du. \]

Simplify and Integrate

Split the integral: \[ \int_{1/2}^{\sqrt{3}/2} (\frac{1}{\sqrt{u}} - u^{3/2}) \, du.\]Evaluate each part separately:\[\int (1/u^{1/2}) \, du = 2u^{1/2}, \quad \int u^{3/2} \, du = \frac{2}{5}u^{5/2}. \]

Evaluate the Definite Integrals

Compute each integral from \( 1/2 \) to \( \sqrt{3}/2 \):- For \( 2u^{1/2} \): \[ 2(\sqrt{3}/2)^{1/2} - 2(1/2)^{1/2} = 2(\frac{2}{\sqrt{3}} - \frac{1}{\sqrt{2}}). \]- For \( \frac{2}{5}u^{5/2} \): \[ \frac{2}{5}((\sqrt{3}/2)^{5/2} - (1/2)^{5/2}) = \frac{2}{5}(\frac{3\sqrt{3}}{4} - \frac{1}{4\sqrt{2}}). \] Add the results together.

Calculate the Numeric Result

After substituting the limits into each part, evaluate them separately and then combine to obtain the numeric result, as required by the problem statement. The numerical computation yields approximately 0.569 after rounding to three decimal places.

Key concepts

Trigonometric Substitution

Understanding trigonometric substitution is crucial for solving integrals that involve trigonometric functions, such as in this exercise. Here, we can use a trigonometric identity to transform the integral into a more manageable form. By rewriting \( \cos^3 x \) using the identity \( \cos^2 x = 1 - \sin^2 x \), we simplify our expression: \( \cos^3 x = (1 - \sin^2 x) \cos x \). This restructuring allows us to use substitution techniques effectively.
Additionally, substituting \( u = \sin x \) and thus \( du = \cos x \, dx \), modifies the original variable \( x \) in our integral to \( u \), which greatly simplifies the integration process. With changed limits, this substitution converts the integral into a simpler form: \( \int_{1/2}^{\sqrt{3}/2} \frac{1 - u^2}{\sqrt{u}} \, du \). This key step leveraging trigonometric relationships enables us to handle otherwise complex integrals more easily.

Integration Techniques

Integration techniques are systematic processes to find the integral of a function. In our problem, we've combined substitution and splitting of integrals. Substitution is pivotal because it transforms a challenging integral into a more straightforward form. By substituting \( u = \sin x \), the problem shifts into a polynomial function of \( u \), which is easier to integrate.
After substitution, another technique is applied: splitting the integral into parts that can be solved separately: \( \int \left(\frac{1}{\sqrt{u}} - u^{3/2}\right) \, du \). This allows us to handle each part with simpler, more direct integration techniques. The integral \( \int \frac{1}{u^{1/2}} \, du \) evaluates to \( 2u^{1/2} \), while \( \int u^{3/2} \, du \) evaluates to \( \frac{2}{5}u^{5/2} \). These techniques together streamline the solution process and illustrate the flexibility of integration methods.

Exact Form

Expressing integrals in exact form means representing the answer in terms of a precise mathematical expression, rather than a numerical approximation. In the realm of integrals, exact forms often involve radicals, constants, and other algebraic expressions that provide a deeper understanding of the function's behavior across the interval.
In this exercise, after integrating and substituting back the variable, exact form involves radicals that inherently appear from functions such as \( 2(\sqrt{3}/2)^{1/2} \) and \( 2(1/2)^{1/2} \). These forms give a clear, concise mathematical expression of the result, offering insights into the nature of the function without approximations. Exact forms are often preferable in mathematics when precision and expression in terms of known functions are vital.

Numerical Approximation

Numerical approximation is a vital concept in calculus for estimating the value of integrals when exact forms are cumbersome or impossible to compute by hand. Even though we used established integration techniques to find the integral's exact form, the problem asks for a numerical approximation, showing the practical applicability of approximation methods.
To achieve an approximation, the calculated exact form is evaluated using numerical methods or calculators to yield a decimal value. In this instance, after careful evaluations, the numerical result is approximately 0.569 when rounded to three decimal places. This approximation is important for practical applications where precise numerical results are needed rather than algebraic expressions.