Problem 53 - Applied Calculus

Question

Find the indefinite integrals. $$ \int \sin t d t $$

Step-by-Step Solution

Answer

\( \int \sin t \, dt = -\cos t + C \)

Step 1: Identify the Integral

The exercise asks us to find the indefinite integral of the function \( \sin t \). This means we need to determine the antiderivative of \( \sin t \).

Step 2: Recall the Antiderivative Rule for Sine Function

The antiderivative of \( \sin t \) is derived from the basic trigonometric integral rules. Recall that the derivative of \( -\cos t \) is \( \sin t \). Thus, the integral of \( \sin t \) must be \( -\cos t \).

Step 3: Include the Constant of Integration

An indefinite integral will always include a constant of integration, denoted as \( C \), to account for all possible antiderivatives. Therefore, the integral of \( \sin t \) is expressed as: \[ \int \sin t \, dt = -\cos t + C \]

Step 4: Finalize Result

The final indefinite integral of \( \sin t \) is succinctly given by: \[ -\cos t + C \] This completes the solution to the integral.

Key Concepts

Antiderivative

Finding an antiderivative is like working backwards from differentiation. The goal is to find a function whose derivative matches the given function. In this exercise, you're given the function \( \sin t \) and need to find its antiderivative. Since the derivative of \( -\cos t \) is \( \sin t \), it's clear that \( -\cos t \) is an antiderivative of \( \sin t \). This backward process helps us understand the indefinite integral process.

Remember, while differentiation is unique—different functions can yield the same derivative—a function can have multiple antiderivatives. That's why when calculating an indefinite integral, more than one solution is possible, each differing by a constant.

Trigonometric Integrals

Trigonometric integrals are a vital part of integral calculus that focus on integrating functions involving trigonometric expressions. A common example is integrating functions like \( \sin t \) or \( \cos t \).

To find the integral of \( \sin t \), we look back to a core rule: the derivative of \( -\cos t \) is \( \sin t \). Therefore, the integral \( \int \sin t \, dt \) is \( -\cos t \). Integrals involving trigonometric functions are often solved by recalling such derivative rules.

Key points for trigonometric integrals include:

• Recognizing standard derivatives of trigonometric functions.
• Understanding relationships between sine and cosine functions.
• Using substitution and identities to simplify when needed.

Approach these integrals methodically and remember the fundamental trigonometric identities as they often make the process smoother.

Constant of Integration

The constant of integration is crucial when calculating indefinite integrals because it acknowledges the possibility of different antiderivatives. Whenever you find an indefinite integral, you must add this constant \( C \) to your result.

This constant arises because antiderivatives are not unique. For example, both \( -\cos t + 1 \) and \( -\cos t + 5 \) have the same derivative \( \sin t \). Therefore, to reflect the infinite family of possible antiderivatives, we include a \( C \) in every indefinite integral solution.

Key ideas around the constant of integration include:

• Always add \( C \) to your indefinite integral results.
• \( C \) can represent any real number, revealing each different antiderivative.
• In definite integrals, the constant cancels out, so it doesn't appear in those results.

Understanding the constant of integration ensures accuracy and completeness in integral calculus problems.