Problem 52 - Calculus Single Variable

Question

Use the Chain Rule to calculate the given indefinite integral. $$ \int \cos (\sin (x)) \cos (x) d x $$

Step-by-Step Solution

Answer

\( \int \cos (\sin (x)) \cos (x) \, dx = \sin(\sin(x)) + C \)

Step 1: Identify the Inner Function

The integral involves a composition function, where \( u = \sin(x) \). Recognizing this helps in applying the chain rule effectively.

Step 2: Differentiate the Inner Function

Calculate the derivative of the inner function: \( \frac{du}{dx} = \cos(x) \). This will be used for substitution.

Step 3: Substitute and Simplify the Integral

Use the substitution \( u = \sin(x) \), and \( du = \cos(x) \, dx \). The integral becomes \( \int \cos(u) \, du \).

Step 4: Integrate with Respect to u

The integral \( \int \cos(u) \, du \) is straightforward and equals \( \sin(u) + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back the Original Variable

Replace \( u \) with \( \sin(x) \) from our initial substitution. Therefore, \( \sin(u) + C \) becomes \( \sin(\sin(x)) + C \).

Key Concepts

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus, representing the reverse process of differentiation. When you integrate a function indefinitely, you find a family of functions whose derivative gives the original integrand. The result of an indefinite integral is called the antiderivative and is expressed along with a constant of integration, usually denoted by \(C\). This constant is essential because differentiation of a constant results in zero, and hence different functions may differ by a constant yet have the same derivative.
For example:

• When you integrate \(f(x) = K\), you find all functions \(F(x) + C\) where \(F'(x) = f(x)\).
• In the integral \( \int \cos (\sin (x)) \cos (x) \, dx \), the goal is to find such a function whose derivative is \( \cos (\sin (x)) \cos (x) \).

Understanding indefinite integrals is a crucial step before tackling more complex problems like definite integrals or specific applications where initial conditions are involved.

Substitution Method

The substitution method is a powerful technique for solving integrals, especially when dealing with composite functions. It involves using a substitution to simplify the integral, often transforming it into a more manageable form. This method is grounded in the chain rule of differentiation and is particularly effective when an integrand contains a function and its derivative.
Here are some steps to follow when using the substitution method:

• Choose the Substitution: Identify the inner function that, when substituted, will simplify the integral. In our original problem, let \( u = \sin(x) \).
• Differentiate to Find \( du \): Find the differential of the substitution \( u \). The derivative of \( u = \sin(x) \) is \( du = \cos(x) \, dx \).
• Rewrite the Integral: Replace the variables in the integral with \( u \) and \( du \). The integral \( \int \cos (\sin (x)) \cos (x) \, dx \) becomes \( \int \cos(u) \, du \).
• Integrate Using Simpler Form: Solve the integral in terms of \( u \), and then substitute back to the original variable to get the final solution.

By mastering the substitution method, you can tackle a wide array of integrals that initially seem complex.

Derivative of Trigonometric Functions

The derivative of trigonometric functions is a vital aspect of calculus, providing insight into the rate of change of these periodic functions. Understanding these derivatives is key to solving not only integrals but various mathematical problems involving oscillatory motion or phenomena.
Key derivatives that you should be familiar with include:

• \( \frac{d}{dx} \sin(x) = \cos(x) \)
• \( \frac{d}{dx} \cos(x) = -\sin(x) \)
• \( \frac{d}{dx} \tan(x) = \sec^2(x) \)

In the exercise provided, the derivative of \( \sin(x) \) is \( \cos(x) \), which plays a critical role in simplifying the integral via substitution.
Recognizing these derivatives and being able to apply them correctly is essential, especially when the integrand contains products or compositions involving trigonometric functions. When integrating, sometimes you'll find yourself needing to recognize a pattern that looks like a derivative of a trigonometric identity, which can simplify the process substantially.