Problem 22 - Thomas Calculus in SI Units

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. $$\int\left(1-x^{2}-3 x^{5}\right) d x$$

Step-by-Step Solution

Answer

The antiderivative is \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\).

Step 1: Identify the Function Components

The function to integrate is: \(1 - x^2 - 3x^5\). We need to find the antiderivative of each term separately.

Step 2: Integrate Each Term

Integrate each term of the function:1. The integral of \(1\) is \(x\).2. The integral of \(-x^2\) is \(-\frac{x^3}{3}\).3. The integral of \(-3x^5\) is \(-\frac{3x^6}{6} = -\frac{x^6}{2}\).

Step 3: Combine the Integrals

Combine the integrals to get the most general antiderivative:\[\int (1 - x^2 - 3x^5) \, dx = x - \frac{x^3}{3} - \frac{x^6}{2} + C\]where \(C\) is the constant of integration.

Step 4: Verify by Differentiation

Differentiate the obtained antiderivative:\[\frac{d}{dx} \left(x - \frac{x^3}{3} - \frac{x^6}{2} + C\right) = 1 - x^2 - 3x^5\].This matches the original function, confirming our solution is correct.

Key Concepts

Antiderivative

The concept of an antiderivative is foundational in calculus, often intertwined with the idea of integration. When we search for an antiderivative of a function, we are essentially looking for another function whose derivative is the given function. This reverse process of differentiation allows us to find the original function before it was differentiated. In simpler terms, if you have a function and you know its derivative, the antiderivative is a function that will give you back the original function when you take its derivative.To find an antiderivative of a polynomial function, such as the one in our original exercise, we integrate each term separately:

• For constant terms, integrate directly to a multiple of the variable, like the integral of 1 becomes x.
• For terms like \(x^n\), the antiderivative is \(rac{x^{n+1}}{n+1}\), adjusting the exponent and dividing by the new exponent value.

Combining these, we find the most general antiderivative, which forms the foundation of the process called indefinite integration. This gives us a comprehensive function plus a constant of integration, which represents the family of all possible antiderivatives.

Differentiation

Differentiation is the process of finding a derivative, which tells us how a function changes as its input changes. It's a key operation in calculus, providing the rate of change or slope of a curve at any given point. In solving integrals, differentiation serves as a verification step to ensure our antiderivative was found correctly.For example, in the given exercise, after finding the antiderivative \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\), we differentiate it to check against the original function. This involves applying power rules in reverse:

• The derivative of \(x\) is \(1\).
• For \( -\frac{x^3}{3}\), the derivative brings down the power, yielding \(-x^2\).
• The term \( -\frac{x^6}{2}\) when differentiated results in \(-3x^5\).
• The constant \(C\) differentiates to zero as it does not change with respect to x.

Differentiation helps confirm that the antiderivative aligns with its derivative, reinforcing the correctness of our solution.

Constant of Integration

The constant of integration, denoted as \(C\), is an essential part of any indefinite integral. This constant represents the family of solutions for the antiderivative. When integrating a function, different antiderivatives can produce a range of functions which differ by a constant. Therefore, the constant of integration accounts for these differences.Imagine the process of differentiation: differentiating a constant results in zero. Hence, when finding the antiderivative, it's impossible to determine this lost constant without additional information. Adding \(C\) ensures that our solution is as general as possible, encompassing all potential original functions that could have produced the derivative.In practice, when solving an indefinite integral like \(\int (1 - x^2 - 3x^5) \, dx\), we write the result as \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\), embracing the infinite possibilities of functions differing by a constant. This constant is crucial in fields that require precise calculations, such as physics and engineering, where specific initial conditions help determine the particular solution from this family.