Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               
                                                                                                                                                                  
 (d) ∫ cos 𝓍/7 d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves a cosine function divided by a constant. The general formula for the integral of cos(𝓍) is ∫cos(𝓍)d𝓍 = sin(𝓍) + C, where C is the constant of integration.
2. Step 2: Notice that the argument of the cosine function is scaled by a constant factor (1/7). To handle this, use the substitution rule for integrals. Let u = 𝓍/7, which implies du = (1/7)d𝓍.
3. Step 3: Rewrite the integral in terms of u. Substituting u = 𝓍/7 and du = (1/7)d𝓍, the integral becomes ∫cos(u) * 7 du.
4. Step 4: Apply the formula for the integral of cos(u). The integral of cos(u) is sin(u), so the integral becomes 7 * sin(u) + C.
5. Step 5: Substitute back u = 𝓍/7 to express the result in terms of the original variable. The final expression is 7 * sin(𝓍/7) + C.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with the integral sign followed by the function and the differential, and they include a constant of integration (C) since the derivative of a constant is zero. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they are foundational for determining areas under curves and solving differential equations.

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be straightforward. Common techniques include substitution, integration by parts, and using integral tables. In this context, referring to Table 5.6 suggests that specific integrals have been pre-calculated, allowing students to apply these results directly to evaluate the given integral without performing the entire calculation from scratch.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and are often encountered in integration problems. The integral of cosine, for example, is a common form that can be evaluated using known results or integration techniques. Understanding the properties and graphs of these functions is essential for recognizing patterns and applying the appropriate integration methods effectively.