Question

Perform the indicated integrations. $$ \int \sin ^{5} 4 x \cos ^{2} 4 x d x $$

Short answer

\(-\frac{1}{4} \left( \frac{\cos^3(4x)}{3} - \frac{2\cos^5(4x)}{5} + \frac{\cos^7(4x)}{7} \right) + C\)

Step-by-step solution

Simplify the Integral Using Trigonometric Identity

First, we need to simplify the integral by using a trigonometric identity. Notice that \( \sin^5(4x) = (\sin^4(4x))\sin(4x) = (\sin^2(4x))^2 \sin(4x) \). Using the identity \( \sin^2(4x) = 1 - \cos^2(4x) \), rewrite \( \sin^5(4x) \) in terms of \( \sin(4x) \) and \( \cos(4x) \) as:\[\sin^5(4x) = (1 - \cos^2(4x))^2 \sin(4x)\]

Substitute Simplified Expression into Integral

Substitute the simplified expression for \( \sin^5(4x) \) into the integral:\[\int (1 - \cos^2(4x))^2 \sin(4x) \cos^2(4x) \, dx\]

Use Substitution Method

To simplify the integral, use the substitution method. Let \( u = \cos(4x) \), then \( du = -4 \sin(4x) \, dx \), or equivalently, \( \sin(4x) \, dx = -\frac{1}{4} \, du \). Substitute these into the integral:\[-\frac{1}{4} \int (1 - u^2)^2 u^2 \, du\]

Expand and Integrate with Respect to u

Expand \((1 - u^2)^2\) and substitute it into the integral:\[(1 - u^2)^2 = 1 - 2u^2 + u^4\]Substituting this, you get:\[-\frac{1}{4} \int (1 - 2u^2 + u^4) u^2 \, du = -\frac{1}{4} \int (u^2 - 2u^4 + u^6) \, du\]Integrate each term separately:\[-\frac{1}{4} \left( \frac{u^3}{3} - \frac{2u^5}{5} + \frac{u^7}{7} \right) + C\]

Substitute Back to Original Variable

Replace \(u = \cos(4x)\) back into the integrated result to express it in terms of \(x\):\[-\frac{1}{4} \left( \frac{\cos^3(4x)}{3} - \frac{2\cos^5(4x)}{5} + \frac{\cos^7(4x)}{7} \right) + C\]

Final Result

The complete integration, after substituting back, gives us:\[-\frac{1}{4} \left( \frac{\cos^3(4x)}{3} - \frac{2\cos^5(4x)}{5} + \frac{\cos^7(4x)}{7} \right) + C\] This is the final result of the integration.

Key concepts

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying complex integrals involving trigonometric functions. They help transform the expressions into simpler forms that are easier to integrate. For instance, in the given problem, we have the function \( \sin^5(4x) \). By using the identity \( \sin^2(x) = 1 - \cos^2(x) \), we can rewrite \( \sin^5(4x) = (\sin^2(4x))^2 \sin(4x) \) as \((1 - \cos^2(4x))^2 \sin(4x)\).
This transformation significantly simplifies the process of integration by reducing the powers involved. It's important to become familiar with a variety of trigonometric identities, such as the Pythagorean identities:

• \( \sin^2(x) + \cos^2(x) = 1 \)
• \( 1 + \tan^2(x) = \sec^2(x) \)
• \( 1 + \cot^2(x) = \csc^2(x) \)

These identities are powerful tools for breaking down trigonometric functions into simpler parts.

Substitution Method

The substitution method is a common technique in integral calculus used to simplify integrals, particularly when the integral is not easily solvable in its original form. In the exercise, after rewriting the integral in terms of \( \cos(4x) \), the substitution \( u = \cos(4x) \) was introduced.
This step involves differentiating \( u \) with respect to \( x \) to find \( du \). We find that \( du = -4 \sin(4x) \, dx \), which simplifies to \( \sin(4x) \, dx = -\frac{1}{4} \, du \). This substitution converts the integral into a simpler polynomial form of \( u \).
Such transformations make the integration process more manageable by reducing the complexity of the trigonometric functions involved. Substitution is incredibly effective whenever the integral involves a chain of functions, enabling you to integrate without directly handling complex compositions.

Integral Calculus

Integral calculus focuses on the concept of integrating functions, essentially finding the area under the curve of a given function. In this exercise, after substitution, the integral was transformed into \[-\frac{1}{4} \int (1 - u^2)^2 u^2 \, du\].
By expanding and simplifying the expression, it becomes possible to integrate each term independently:

• \(-\frac{1}{4} \int u^2 \, du\)
• \(-2 \int u^4 \, du\)
• \(\int u^6 \, du\)

The rule for integrating \( u^n \) is to use the power rule, which states that \( \int u^n du = \frac{u^{n+1}}{n+1} + C \).
This approach allows us to tackle integrals systematically, solving each term individually and then combining them to achieve the full solution.

Definite Integrals

Definite integrals are integral expressions evaluated over a specified interval, giving the net area under a curve. In contrast, the exercise we examined is an indefinite integral, which does not have specific limits of integration.
While definite integrals result in a numerical value representing the total accumulated quantity over an interval, indefinite integrals represent a family of functions, with '+ C' indicating an arbitrary constant.
Understanding this distinction is key in integral calculus, as it guides when to apply the evaluated boundary limits and when to generalize the solution across an infinite set of possibilities. Both definite and indefinite integrals are cornerstones in calculus, each serving unique and critical roles.