Problem 22 - 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. $$\int\left(1-x^{2}-3 x^{5}\right) d x$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Integral Expression

The integral given is \( \int (1 - x^2 - 3x^5) \, dx \). We are tasked with finding the general antiderivative of this expression. Each term in the polynomial will be integrated separately.

Step 2: Integrate Each Term Separately

To find the antiderivative, integrate each term of the expression separately:1. The integral of \(1\) is \(x\), since \(d(x) = dx\).2. The integral of \(-x^2\) is \(-\frac{x^3}{3}\).3. The integral of \(-3x^5\) is \(-\frac{3x^6}{6}\), which simplifies to \(-\frac{x^6}{2}\).

Step 3: Combine the Integrals

Add the antiderivatives together, including the constant of integration \(C\):\[\int (1 - x^2 - 3x^5) \, dx = x - \frac{x^3}{3} - \frac{x^6}{2} + C\]

Step 4: Verify by Differentiation

Differentiate the result \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\) to confirm it equals the original integrand:1. The derivative of \(x\) is \(1\).2. The derivative of \(-\frac{x^3}{3}\) is \(-x^2\).3. The derivative of \(-\frac{x^6}{2}\) is \(-3x^5\).4. The derivative of \(C\) is \(0\).Combine these to get \(1 - x^2 - 3x^5\), confirming the original integrand.

Key Concepts

Antiderivative

An antiderivative of a function is essentially the reverse process of differentiation. It is a function whose derivative is equal to the original function you started with. When we talk about finding the antiderivative, we are really looking for a function that, when differentiated, gives us the function under the integral sign, also called the integrand.
For the exercise \(\int (1 - x^2 - 3x^5) \, dx\), we want to find the function that, when differentiated, returns \(1 - x^2 - 3x^5\). This involves integrating each term within the polynomial separately and making sure to include a constant of integration, which represents any constant value added to our function without influencing its derivative.

Polynomial Integration

Polynomial integration is a straightforward technique involving integrating terms in a polynomial function one by one by increasing their power by one and dividing by the new power. It is one of the basic operations in calculus and serves as a fundamental building block for solving more complex integrals.
To integrate polynomials like \(1 - x^2 - 3x^5\), break it down term by term:

• The integral of \(1\) is \(x\), because the derivative of \(x\) is \(1\).
• For \(-x^2\), increase the exponent by one to get \(-x^3\) and divide by this new power, giving \(-\frac{x^3}{3}\).
• Finally, for \(-3x^5\), the integral becomes \(-\frac{3x^6}{6}\) which simplifies to \(-\frac{x^6}{2}\).

Combine these integrated terms, and you have the antiderivative for the polynomial expression.

Constant of Integration

When finding an indefinite integral, a constant of integration \(C\) is added to represent the family of antiderivatives. This is because differentiation of a constant is zero, and any constant added will vanish when taking the derivative.
In our solution to \(\int (1 - x^2 - 3x^5) \, dx = x - \frac{x^3}{3} - \frac{x^6}{2} + C\), the \(C\) ensures that all possible vertical shifts of the antiderivative are included. Without this constant, the solution would only describe a single function rather than the whole set of functions fitting the integral's derivative.

Differentiation Check

After computing an antiderivative, it's crucial to check the work by differentiating the result. This step ensures accuracy by confirming whether the derived function returns to the original integrand.
For the antiderivative \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\), differentiate each term:

• Derivative of \(x\) is \(1\).
• Derivative of \(-\frac{x^3}{3}\) is \(-x^2\).
• Derivative of \(-\frac{x^6}{2}\) is \(-3x^5\).
• Derivative of \(C\) is \(0\).

Adding these results returns the original function \(1 - x^2 - 3x^5\), verifying the correctness of our integration.