Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-28, perform the indicated integrations. \(\int \sin ^{1 / 2} 2 z \cos ^{3} 2 z d z\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{1}{3}(\sin(2z))^{3/2} - \frac{1}{5}(\sin(2z))^{5/2} + C\).

Step 1: Use Trigonometric Identities

In order to solve the integration problem, we can begin by using a trigonometric identity that simplifies products of sine and cosine functions. Recall the identity: \[\sin^{a}x \cos^{b}x = \left(\sin^{2}x\right)^{a/2}\left(\cos^{2}x\right)^{b/2}\sin(x)\cos(x)\]Applying this to \(\sin^{1/2}(2z)\cos^{3}(2z)\), noting that \(\sin(2z)\cos(2z) = \frac{1}{2}\sin(4z)\):\[\frac{1}{2}\sin^{1/2}(2z)\left(\cos^{4}(2z)\right)^{1/2}\sin(4z)\] creates a different integral form.

Step 2: Substitute with a Trigonometric Substitution

From Step 1, we simplify further by noting that \(\sin^{1/2}(2z) = \sqrt{\sin(2z)}\) and \(\cos^{3}(2z)\cos(2z) = \cos^{2}(2z)\cos(2z)\). We simplify using the substitution \( u = \sin(2z) \), then \( du = 2\cos(2z) dz \), or \( \frac{1}{2}du = \cos(2z) dz\). The integral becomes:\[\int \sqrt{u} \left(1-u\right)\frac{1}{2}du\] which sets up for use of substitution effectively.

Step 3: Simplify and Integrate with Respect to u

Now, integrate the simplified integral: \[\frac{1}{2}\int \sqrt{u} \left(1-u\right) du\].Distribute inside the integrand:\[\frac{1}{2}\int(u^{1/2} - u^{3/2})du \].Integrate each term separately:\[\frac{1}{2}\left(\frac{u^{3/2}}{3/2} - \frac{u^{5/2}}{5/2}\right) + C\] becomes:\[\frac{1}{2}\left(\frac{2}{3}u^{3/2} - \frac{2}{5}u^{5/2}\right) + C\], simplifying to:\[\frac{1}{3}u^{3/2} - \frac{1}{5}u^{5/2} + C\].

Step 4: Back-Substitute and Simplify

Recall that \(u = \sin(2z)\), so back-substitute to obtain the expression in terms of \(z\):\[\frac{1}{3}(\sin(2z))^{3/2} - \frac{1}{5}(\sin(2z))^{5/2} + C\].This concludes the solution to the integral such that \(z\) is expressed in the final answer.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a clever technique used in calculus for making integrals easier to solve. This method comes in handy especially when dealing with integrals involving square roots or quadratic expressions in the form of sine or cosine. The key idea here is to replace a given variable with a trigonometric function, which can simplify the integral drastically.

In our original exercise, we faced an integral with \(\sin^{1/2}(2z)\) and \(\cos^{3}(2z)\). To simplify it, we can use a substitution. Instead of dealing directly with these terms, we substituted \(u = \sin(2z)\), transforming the integral into a simpler problem. This approach leverages the relationship between trigonometric functions and their derivatives. It allows us to convert a potentially complicated integral into an easier one.

Using this substitution, our complex original integral becomes tractable, which is crucial when finding an antiderivative analytically. Substitution simplifies the differential element, as seen when \(du = 2\cos(2z) \,dz\) is replaced with \(\frac{1}{2}du = \cos(2z) \,dz\). This simplifies the integral structure remarkably, making it easier to handle.

Integral Simplification

Integral simplification is an important part of solving integrals, especially when they first appear too complex to handle directly. The initial approach might seem daunting due to the nature of functions like \( \sin^{1/2}(2z) \cos^{3}(2z) \). However, by simplifying these terms, solving the integral becomes much more manageable.

The first step involves rewriting complex expressions using known mathematical identities or substitutions. For example, we convert \( \sin^{1/2}(2z) \) into \( \sqrt{\sin(2z)} \) and break down \( \cos^{3}(2z) \) into parts involving \( \cos(2z) \.\) Then, implementing these simplifications in a strategic order, you attain a form that is more digestible for direct integration.

• We split the integrals, making the overall process easier.
• Use algebraic manipulation to rearrange and simplify expressions.
• Identify opportunities to combine like terms.

After simplification, substitute back into the original variables, creating a final expression that is simpler and more intuitive.

Trigonometric Identities

Understanding trigonometric identities is crucial for solving trigonometric integrals. These identities provide relationships between the basic trigonometric functions, allowing us to simplify and transform expressions, making integration far more manageable.

In the context of our integral exercise, we used identities like \( \sin(2z)\cos(2z) = \frac{1}{2}\sin(4z) \) to rewrite expressions into simpler forms. This enables breaking down the initial complex integrands into a series of more approachable terms.

• Identify and apply relevant identities to simplify expressions.
• Convert products of sines and cosines into sum or difference forms.
• Utilize Pythagorean identities to express one function in terms of another.

By employing these identities, we can transform integrals that seem insurmountable at first into forms that are easier to integrate. Mastery of these identities is not just beneficial but necessary for tackling more advanced integration problems effectively.