Problem 9 - Calculus

Question

Find the indefinite integrels in \(1-20\). \(\int \cos ^{3} 2 x \sin 2 x d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(-\frac{1}{8} \cos^4(2x) + C\)."

Step 1: Identify Substitution

In this integral, we notice a composite function where the derivative of the inner function is present. We have \(\cos^3(2x)\sin(2x)\). The derivative of \(\sin(2x)\) is \(2\cos(2x)\). We will substitute \(u = \cos(2x)\), which implies \(du = -2\sin(2x)dx\) or \(\sin(2x)dx = -\frac{1}{2}du\).

Step 2: Substitute into the Integral

Using the substitution from Step 1, we rewrite the integral in terms of \(u\) as \(-\frac{1}{2} \int u^3 \, du\).

Step 3: Integrate with Respect to u

Find the integral of \(u^3\). The integral of \(u^3\) with respect to \(u\) is \(\frac{u^4}{4}\). So, the integral \(-\frac{1}{2} \int u^3 \, du = -\frac{1}{2} \times \frac{u^4}{4} + C = -\frac{u^4}{8} + C\).

Step 4: Substitute Back in for x

Replace \(u\) with \(\cos(2x)\) to return to the original variable, giving \(-\frac{(\cos(2x))^4}{8} + C\).

Step 5: Simplify the Solution

The expression \(\cos^4(2x)\) needs no further simplification in this context, so the solution to the integral is \(-\frac{1}{8}\cos^4(2x) + C\).

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving integrals, especially when dealing with composite functions. The core idea is to identify a part of the integral that can be substituted with a single variable, typically denoted as \( u \). This simplification of the integral often makes it much easier to solve.

In cases where the derivative of the inner function is present in the integral, substitution can be particularly effective. Here's how it's generally done:

• Identify the inner function, \( g(x) \), in your integral.
• Substitute \( u = g(x) \). Calculate \( du = g'(x)dx \).
• Rearrange this expression to substitute \( dx \) in terms of \( du \).
• Transform your integral entirely in terms of \( u \) and \( du \).
• Integrate with respect to \( u \).
• Finally, substitute back the original variable \( x \) using \( u = g(x) \).

This method simplifies handling integrals that at first glance might seem complex and helps gain a clearer view of the problem.

Integration of Composite Functions

When facing the integration of composite functions, understanding the structure of the function is key. Composite functions are made of an inner function nested within an outer function, such as \( f(g(x)) \). To integrate them effectively, one needs to:

• Recognize the composite nature: Look for functions within functions, like trigonometric functions having their argument multiplied or added to other values.
• Use substitution to isolate the inner function: This transforms a complex problem into a straightforward integral.
• Simplify wherever possible: Break down steps to make each smaller piece more manageable.

For instance, in the original exercise, we have \( \cos^3(2x)\sin(2x) \). Recognizing that \( \sin(2x) \) is related to \( \cos(2x) \) through its derivative guided us to substitute \( u = \cos(2x) \). This broke the problem into simpler, manageable parts that were easier to integrate.

Trigonometric Integrals

Trigonometric integrals often involve integrating products or powers of trigonometric functions such as sine and cosine. These integrals can be tricky without some understanding of their properties and relationships.

• Look for derivatives of standard trigonometric identities: This can simplify the integral into a recognizable form.
• Simplify and rewrite the product if possible: Use trigonometric identities to make the integral easier to tackle.
• Apply substitution judiciously: When one trigonometric function is the derivative of another, substitution can simplify the process immensely.

In the exercise at hand, knowing the derivative relationships between \( \sin(2x) \) and \( \cos(2x) \) allowed for an efficient substitution. This transformed the integration of a trigonometric product into a simpler algebraic form, leading to a straightforward solution. Keeping these techniques in mind can significantly ease the process of working with trigonometric integrals.