Problem 36 - Thomas Calculus

Question

In Exercises \(17-56,\) 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 to integrate

The function given is \(-5 \sin t\). Our goal is to find an antiderivative of this function.

Step 2: Find the antiderivative of the sine function

The antiderivative of \(-\sin t\) is \(\cos t\), since the derivative of \(\cos t\) is \(-\sin t\). Therefore, the antiderivative of \(-5 \sin t\) is \(5 \cos t\).

Step 3: Include the constant of integration

Since we are finding the most general antiderivative, we need to include a constant of integration, \(C\). Thus, the general antiderivative is \(5 \cos t + C\).

Step 4: Differentiate to check the solution

To verify, differentiate \(5 \cos t + C\). The derivative of \(5 \cos t\) is \(-5 \sin t\) and the derivative of \(C\) is \(0\). Therefore, the derivative is \(-5 \sin t\), which matches the original function. This confirms our solution is correct.

Key Concepts

Indefinite Integral

The indefinite integral, often referred to as an antiderivative, is the reverse process of differentiation. When you take the indefinite integral of a function, you are essentially finding all functions whose derivative is the original function you started with. You can think of it as "undoing" the derivative.

The notation for an indefinite integral involves the integral sign followed by the function and the differential, for example, \( \int f(t) \, dt \). This operation will yield a new function plus a constant (which we'll discuss shortly).

• To compute an indefinite integral, identify the function you wish to integrate.
• Find a function whose derivative is the given function.
• Don't forget to consider the constant of integration, since indefinite integrals give us a family of functions.

In our exercise, we found the indefinite integral of \(-5 \sin t\) by identifying that the antiderivative involves an element that will differentiate back to the original function.

Constant of Integration

When determining an indefinite integral, we often encounter an element called the "constant of integration". This is represented by the symbol \( C \). It’s important because the differentiation of a constant is always zero, meaning different functions can have the same derivative if they only differ by a constant.

Including \( C \) ensures that we account for all possible functions. This is why when you see \( \int f(t) \, dt = F(t) + C \), the \( C \) is essential as it represents an entire set of vertical shifts of the function \( F(t) \), all of which have the derivative \( f(t) \).

In our solution for the integral \( \int (-5 \sin t) \, dt \), the general antiderivative is given as \( 5 \cos t + C \). This expresses that any function of the form \( 5 \cos t \) plus a constant will have a derivative of \(-5 \sin t\).

Differentiation

Differentiation is the mathematical process of finding the derivative of a function. The derivative represents the rate of change of a function at any given point and is a fundamental concept in calculus.

When it comes to validating the result of an indefinite integral, differentiation can be used to check if the computed antiderivative is correct. Simply differentiate the result and see if it returns the original function. If it does, the antiderivative is verified as correct.

• Consider the function you obtained from integrating—let's say \( F(t) + C \).
• Differentiate \( F(t) \). The derivative of \( C \), being a constant, is zero.
• If the derivative matches the original integrand, verification is successful.

In our problem, by differentiating \( 5 \cos t + C \), we found \( -5 \sin t \). This matched the original function, confirming that our integration process was accurate.