Problem 35 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int(-2 \cos t) d t $$

Step-by-Step Solution

Answer

The most general antiderivative is \(-2 \sin t + C\).

Step 1: Identify the Antiderivative Formula

The integral to solve is \( \int (-2 \cos t) \, dt \). The antiderivative of \( \cos t \) is \( \sin t \). So, we will use this fact to find the antiderivative of \(-2 \cos t\).

Step 2: Compute the Antiderivative

Apply the constant factor rule of integration, which states that \( \int c \cdot f(t) \, dt = c \cdot \int f(t) \, dt \), where \( c \) is a constant. Thus, \( \int (-2 \cos t) \, dt = -2 \cdot \int \cos t \, dt = -2 \sin t + C \), where \( C \) is the constant of integration.

Step 3: Differentiate to Check the Solution

To verify the result, differentiate \(-2 \sin t + C\). The derivative of \( \sin t \) is \( \cos t \), and the derivative of a constant \( C \) is 0. Therefore, \( \frac{d}{dt}(-2 \sin t + C) = -2 \cos t \). This matches the original integrand, confirming the antiderivative.

Key Concepts

Antiderivative

An antiderivative is essentially the reverse process of differentiation. When we look to find the antiderivative, we're often looking for the function that, when differentiated, gives us the original function inside the integral. So, if you have a function and you want to find its antiderivative, you're finding a function whose derivative matches the original function you started with.

In the problem above, the antiderivative of \(-2 \cos t\) is sought. The antiderivative of \(\cos t\) is \(\sin t\). Therefore, we apply the constant multiple rule of integration, which tells us that we can take constants out of the integral. Thus, the antiderivative of \(-2 \cos t\) becomes \(-2 \sin t + C\), where \(C\) represents the constant of integration. To verify, you can differentiate the antiderivative, and if it returns the original function, then the calculations are correct.

Cosine Function

The cosine function (\(\cos t\)) is a fundamental trigonometric function. It's often featured in calculus problems involving integration and differentiation due to its periodic nature. Understanding this function is crucial because it often appears in physics and engineering problems where wave patterns, circular motion, or harmonic oscillators are involved.

The integral of the cosine function, as seen in the problem above, can be easily identified because it's well-known among fundamental integrals. When integrating \(\cos t\), the result is \(\sin t\). This is because the derivative of \(\sin t\) is \(\cos t\). Thus, finding the antiderivative of \(-2 \cos t\) simply involves applying this knowledge alongside the constant factor rule.

Constant of Integration

The constant of integration, often denoted as \(C\), is an arbitrary constant added to the antiderivative when computing indefinite integrals. It's essential because differentiating a constant returns zero; therefore, when you take the derivative of an antiderivative, the constant "disappears" and you regain the original function.

• Any particular antiderivative may not capture every possible original function.
• Adding \(C\) accounts for all shifted functions that have exactly the same rate of change as the function being integrated.

In the given exercise \(\int(-2 \cos t) dt\), the constant of integration \(C\) ensures that all potential solutions are represented, as there are infinitely many functions that could have the same derivative. When the solution is differentiated, \(C\) vanishes, perfectly confirming that \(-2 \cos t\) is the derivative of \(-2 \sin t + C\). This safeguards against missing out on any "hidden" values that originated from initial conditions or specific contexts in applied problems.

Thus, whenever you calculate an indefinite integral, remember to include the constant of integration to represent a complete family of solutions.