Problem 44 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(-5 \sin t) d t$$

Step-by-Step Solution

Answer

The most general antiderivative is \(5 \cos t + C\).

Step 1: Identify the Function Form

The function to integrate is \(-5 \sin t\). This is a constant multiplied by a basic trigonometric function \(\sin t\). To solve this, we'll use the antiderivative formula for \(\sin t\), which is \(-\cos t\).

Step 2: Apply the Constant Multiple Rule

The constant multiple rule of integration states that the integral of a constant multiplied by a function is the constant multiplied by the integral of the function. Thus, we have: \[\int (-5 \sin t) \, dt = -5 \int \sin t \, dt\]

Step 3: Find the Antiderivative

Using the standard integral \(\int \sin t \, dt = -\cos t\), we can integrate: \[\int (-5 \sin t) \, dt = -5 (-\cos t) = 5 \cos t\]

Step 4: Add the Constant of Integration

The general antiderivative includes a constant of integration, \(C\). Therefore, the most general antiderivative is: \[5 \cos t + C\] where \(C\) is any constant.

Step 5: Check by Differentiating

Differentiate the result \(5 \cos t + C\): \(\frac{d}{dt} [5 \cos t + C] = 5 (-\sin t) = -5 \sin t\). Since differentiating \(5 \cos t + C\) returns the original function \(-5 \sin t\), our solution is verified.

Key Concepts

Indefinite Integral

When you see the term 'indefinite integral,' it refers to finding the antiderivative of a function. An antiderivative is essentially a reverse process of taking a derivative. Unlike definite integrals, indefinite integrals do not have limits of integration.
Instead, they include a constant of integration, usually denoted as \(C\), because the derivative of a constant is zero, so it doesn’t affect the process of differentiation.
Finding an indefinite integral means you are looking for *all possible* functions that can yield the given function as their derivative. In this process, you're often creating a family of functions by incorporating that constant \(C\).

• Indefinite integrals are represented by the symbol \(\int f(t) \ dt\).
• Indefinite integrals are essential in finding the general solutions to differential equations.
• Including \(C\) ensures that you account for all solutions, providing flexibility in finding the specific one that meets any initial conditions.

Constant Multiple Rule

The constant multiple rule is a simplification tool in integration. It states that when you integrate a constant multiplied by a function, you can "pull out" the constant and multiply it with the integral of the function itself.
This makes the process of integration easier because you can deal with complex constants separately and focus on integrating the function.
The rule is mathematically expressed as \(\int k f(t) \, dt = k \int f(t) \, dt\), where \(k\) is a constant.

• This rule saves time by avoiding unnecessary multiplication steps during integration.
• Example: With \(-5\sin t\), you treat \(-5\) as separate and integrate \(\sin t\).
• It's crucial to remember that this rule simplifies the handling of constants without altering the underlying functionality of integration.

Trigonometric Integration

Trigonometric integration involves integrating functions that include trigonometric functions such as \(\sin\), \(\cos\), \(\tan\), etc. These functions have standard antiderivatives memorized to simplify the integration process.
For example, the integral of \(\sin t\) is \(-\cos t\). Knowing these formulas allows for quick integration without needing to derive them each time.

• Trigonometric integrals might initially look intimidating but breaking down the functions into known integrals can immensely simplify the task.
• Recognizing patterns and standard forms within trig functions help fast-track the integration process.
• Remembering common integrals like \(\int \sin t \, dt = -\cos t\) and \(\int \cos t \, dt = \sin t\) are essential tools for solving problems efficiently.

Antiderivative

An antiderivative of a function is another function whose derivative gives back the original function. Finding antiderivatives is a significant part of calculus as it helps in understanding the accumulation of quantities and solving differential equations.
In our exercise, finding the antiderivative of \(-5 \sin t\) leads to the solution \(5 \cos t + C\). Checking the antiderivative is important by differentiating it to verify the original function is regained.

• Every function has an infinite number of antiderivatives, each differing by a constant \(C\).
• It's a practical check by calculating the derivative to see if it matches the initial function.
• This reverse engineering approach ensures that the solution fits within the defined problem set and satisfies all given criteria.