Problem 52 - Applied Calculus

Question

Find the indefinite integrals. $$ \int \cos \theta d \theta $$

Step-by-Step Solution

Answer

\( \int \cos \theta \, d\theta = \sin \theta + C \).

Step 1: Recall Integration Formula for Cosine

The indefinite integral of the cosine function, \( \cos \theta \), is found using a basic integral formula from calculus. The formula to recall here is \( \int \cos \theta \, d\theta = \sin \theta + C \), where \( C \) is the constant of integration.

Step 2: Apply the Integral Formula

Use the integral formula for cosine to solve the given problem:\[\int \cos \theta \, d\theta = \sin \theta + C.\]This is the result of integrating the cosine function with respect to \( \theta \).

Key Concepts

Integral Calculus

Integral Calculus is a major branch of calculus. It is all about finding the antiderivative or integrating a function.

• When you integrate a function, you are essentially finding the area under its curve.
• There are two main types: definite and indefinite integrals.

The exercise focuses on indefinite integrals. These integrals represent a family of functions and are always paired with a constant, commonly denoted as \(C\). This constant ensures that any vertical shift in the function is accounted for. It is like finding the reverse of differentiation.For example, when integrating \( \cos \theta \,d\theta\), we aim to find a function that, when differentiated, gives us back \( \cos \theta \). We use rules and formulas to find this antiderivative.

Integration by Trigonometric Functions

Trigonometric functions, like sine or cosine, have specific integral rules. These functions often appear in many calculus problems because of their periodic nature.To integrate the cosine function, we use the formula:\[ \int \cos \theta \, d\theta = \sin \theta + C \]

• Here, \(\sin \theta\) is the function whose derivative is \(\cos \theta\).
• The integration of trigonometric functions is often quick if you remember these standard results.

This makes solving problems involving trigonometric integrals straightforward. If you remember these key formulas, your calculus problems around trigonometric functions will become much easier to handle.

Constant of Integration

The constant of integration, represented by \(C\), plays an important part in indefinite integrals. It might seem like a simple addition, but it is crucial for completeness.

• When taking an antiderivative, there can be many functions that differ by a constant. That's why \(C\) is used, to account for all these possibilities.
• Ignoring \(C\) means you miss every other possible vertical shift of the function.

Including the constant of integration ensures that the general solution to the problem is complete. It's good practice to always check and include \(C\) in your final answer for indefinite integrals. So, in our example, \(\int \cos \theta \, d\theta = \sin \theta + C\), \(C\) means we consider all possible versions of the sine curve that could be the original function.