Problem 13 - Applied Calculus

Question

Find each integral. $$ \int \cos \frac{2 \pi(t+20)}{365} d t $$

Step-by-Step Solution

Answer

\( \frac{365}{2\pi} \sin \left( \frac{2\pi(t+20)}{365} \right) + C \).

Step 1: Recognize the Integral Form

We start with the integral \( \int \cos \frac{2 \pi(t+20)}{365} \, dt \). This is a standard cosine integral with a linear function inside the cosine.

Step 2: Identify the Inner Function

The inner function of the cosine is \( u = \frac{2\pi(t+20)}{365} \). If we let \( u \) be this function, we need its derivative with respect to \( t \) to complete the substitution.

Step 3: Calculate the Derivative

Find the derivative of \( u \) with respect to \( t \), which gives \( \frac{du}{dt} = \frac{2\pi}{365} \). This will help us in the substitution process.

Step 4: Rewrite the Differential

Rewrite \( dt \) in terms of \( du \) using the relation \( \frac{du}{dt} = \frac{2\pi}{365} \), resulting in \( dt = \frac{365}{2\pi} \, du \).

Step 5: Substitute in the Integral

Substitute \( u \) and \( dt \) into the integral: \[ \int \cos u \cdot \frac{365}{2\pi} \, du \]. This simplifies to \( \frac{365}{2\pi} \int \cos u \, du \).

Step 6: Integrate the Simplified Integral

Integrate \( \cos u \) with respect to \( u \), which is \( \sin u + C \), where \( C \) is the integration constant.

Step 7: Substitute Back the Inner Function

Replace \( u \) with \( \frac{2\pi(t+20)}{365} \) back into the integral result, yielding the final answer: \( \frac{365}{2\pi} \sin \left( \frac{2\pi(t+20)}{365} \right) + C \).

Key Concepts

Trigonometric Integration

Trigonometric integration involves finding the antiderivatives of trigonometric functions. In this exercise, we are faced with an integral that features the cosine function, \( \int \cos \frac{2 \pi(t+20)}{365} d t \). Understanding this concept is crucial for tackling integrals involving trigonometric expressions. Trigonometric integrals often arise in applications across physics and engineering, such as wave motion and oscillations. Here are a few key pointers to keep in mind:

• Integrals of trigonometric functions such as sine and cosine are standard and often easy to remember. For example, the integral of cosine is simply sine.
• When the argument of the trigonometric function is more complex, such as a linear expression, substitution methods like the one used in this exercise simplify the process.

Recognizing the structure of the integral is the first step in determining the best integration technique to apply. When you identify a trigonometric function, checking for a possible substitution can streamline your solution.

Substitution Method

The substitution method is a clever integration technique used to simplify more complicated integrals by substituting part of the integral with a simpler variable. This technique is particularly useful when working with trigonometric functions that are composed with more complex arguments.In the original exercise, observe the substitution process:

• Identify the \'inner function\' present within the trigonometric function, here \( u = \frac{2\pi(t+20)}{365} \).
• Compute the derivative of the inner function with respect to \( t \), \( \frac{du}{dt} = \frac{2\pi}{365} \), to relate \( dt \) and \( du \).
• Replace expressions in the integral to transform it into a simpler form.

This method reduces the complexity of the original integral by converting it into an equivalent version that's easier to evaluate. Once the integration is performed with respect to \( u \), simply replace \( u \) back with the original expression. This completes the substitution process, making complicated integrals manageable.

Integration Techniques

Integration techniques are varied and tailored to addressing different forms of integrals. Each technique aims to transform the integral to a more familiar form that can be directly integrated. In this context, trigonometric integration and substitution methods were both employed. Here are a few notes on the broader suite of integration techniques:

• Basic Formulas: Learn and remember standard integrals of basic functions, such as polynomials and trigonometric functions.
• Substitution: This is often the first technique to try, especially useful when the integrand includes functional compositions.
• Parts and Partial Fractions: Techniques like integration by parts and partial fraction decomposition can also be extremely beneficial for complex integrals.

By matching the structure of the integral to an appropriate technique, the process of integration becomes more efficient. Understanding these methods and when to apply them is essential for solving integrals successfully.