Problem 68 - Calculus Single Variable

Question

When \(\sin (x)\) and \(\cos (x)\) are both raised to the same positive power in an integrand, the identity \(\sin (2 x)=2 \sin (x) \cos (x)\) may be used to simplify the integral. Use this observation as the basis for calculation of the integrals. $$ \int \cos ^{3}(x) \sin ^{3}(x) d x $$

Step-by-Step Solution

Answer

The integral simplifies to \( \frac{1}{48} \sin^3(2x) + C \).

Step 1: Use Trigonometric Identity

Recognize that both \( \sin(x) \) and \( \cos(x) \) are raised to the same power. Use the identity \( \sin(2x) = 2\sin(x)\cos(x) \) to rewrite the integrand. Begin by expressing \( \cos^3(x)\sin^3(x) \) using this identity: \( \sin^2(x)\cos^2(x) = \left(\frac{1}{4}\sin^2(2x)\right) \).

Step 2: Simplify the Integrand

Rewrite the integrand \( \cos^3(x)\sin^3(x) \) in terms of \( \sin(2x) \). Notice that \( \cos^3(x)\sin^3(x) = \cos(x)\sin(x)(\cos(x)\sin(x))^2 = \frac{1}{4} \cos(2x)\sin^2(2x) \).

Step 3: Substitution Method

Let \( u = \sin(x)\cos(x) \), then \( du = \frac{1}{2}(\cos^2(x) - \sin^2(x))dx = \frac{1}{2}\cos(2x)dx \). Substitute back into the integral: \( \int \frac{1}{4} \cos(2x)\sin^2(2x)dx = \int \frac{1}{4} u^2(2du) \).

Step 4: Integrate

Integrate the function in terms of \( u \): \( \frac{1}{2} \int u^2 \, du = \frac{1}{2} \cdot \frac{u^3}{3} = \frac{1}{6} u^3 \). Substitute back \( u = \sin(x)\cos(x) \) to obtain \( \frac{1}{6} (\sin(x)\cos(x))^3 + C \).

Step 5: Final Simplification

Simplify the expression. Using \( \sin(x)\cos(x) = \frac{1}{2} \sin(2x) \), the integral becomes \( \frac{1}{6} \left( \frac{1}{2} \sin(2x) \right)^3 + C = \frac{1}{48} \sin^3(2x) + C \).

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in calculus, especially for simplifying complex expressions within integrals. These identities are equations involving trigonometric functions that hold true for all values of the variable for which the functions are defined. One famous identity used in integration is \( \sin(2x) = 2\sin(x)\cos(x) \). This particular identity allows us to transform products of sine and cosine functions, particularly when raised to powers, into a more manageable form. In the problem presented, both \( \sin(x) \) and \( \cos(x) \) are raised to the third power. By recognizing this pattern, the identity \( \sin(2x) = 2\sin(x)\cos(x) \) serves as a crucial step in simplifying the integral. This simplification transforms a potentially daunting expression into something more straightforward, thus reducing the complexity of the integral.

Substitution Method

The substitution method is a technique used in integration to simplify an integral, making it easier to evaluate. This is achieved by substituting a part of the integral with a single variable, often denoted by \( u \). The goal of substitution is to convert the integral into a basic form that is easy to integrate. In this specific case, the substitution utilizes the relationship \( u = \sin(x)\cos(x) \). Differentiating with respect to \( x \) gives us \( du = \frac{1}{2}(\cos^2(x) - \sin^2(x))dx = \frac{1}{2}\cos(2x)dx \). Incorporating this substitution into the integral simplifies the complexity, allowing it to be expressed in terms of \( u \). Subsequently, the integral takes on the simpler form \( \int u^2 \, du \), which is straightforward to integrate. The substitution method not only simplifies the process but also illuminates the integral's underlying structure.

Simplification of Integrals

Simplification of integrals is a crucial step in solving complex integration problems, as it transforms difficult expressions into more manageable ones. This process revolves around identifying patterns or identities within the integrand that can lead to easier integration. For the integral \( \int \cos^3(x)\sin^3(x) \, dx \), simplification begins with recognizing the symmetric powers of sine and cosine. By expressing \( \cos(x)\sin(x) \) in terms of \( \sin(2x) \), further substitutions make the integral less intricate. Transforming complex products into powers of simpler trigonometric functions reduces unnecessary complexity. Ultimately, applying both trigonometric identities and substitutions allows for the integral to be expressed in a more intuitive way. This demonstrates the necessity and effectiveness of simplification when tackling intricate trigonometric integrals.