Problem 30 - 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. $$\int\left(1-x^{2}-3 x^{5}\right) d x$$

Step-by-Step Solution

Answer

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

Step 1: Apply the Power Rule for Integration

To find the antiderivative of each term in the expression \( 1 - x^2 - 3x^5 \), we apply the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is an arbitrary constant. Let's integrate each term separately.

Step 2: Integrate the First Term

The first term is 1, which can be expressed as \( x^0 \). Applying the power rule, we get: \( \int 1 \, dx = x + C_1 \).

Step 3: Integrate the Second Term

For the second term \( -x^2 \), apply the power rule: \( \int -x^2 \, dx = -\frac{x^{3}}{3} + C_2 \).

Step 4: Integrate the Third Term

For the third term \( -3x^5 \), use the power rule: \( \int -3x^5 \, dx = -\frac{3x^{6}}{6} + C_3 = -\frac{x^6}{2} + C_3 \).

Step 5: Combine the Antiderivatives

Combine the results of each integration: \( x - \frac{x^3}{3} - \frac{x^6}{2} + C \), where \( C = C_1 + C_2 + C_3 \) is the general constant of integration.

Step 6: Verify by Differentiation

Differentiate \( x - \frac{x^3}{3} - \frac{x^6}{2} + C \) to confirm that it matches the original function. The derivative is \( 1 - x^2 - 3x^5 \), which confirms our solution.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique used to solve indefinite integrals, particularly when dealing with polynomial expressions. The rule simplifies the process of integrating a term of the form \(x^n\) and is given by the formula:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( C \) represents the constant of integration, which accounts for any constant values that disappear when differentiating. Breaking down this process:

• Add one to the exponent \(n\) of the variable \(x\).
• Divide the term by the new exponent \(n+1\).
• Add the constant of integration \(C\).

For example, consider the integral of \(1 - x^2 - 3x^5\):

• The term \(1\) is equivalent to \(x^0\), and its integral is simply \(x\).
• The term \(-x^2\) becomes \(-\frac{x^3}{3}\) after integration.
• The term \(-3x^5\) simplifies to \(-\frac{x^6}{2}\) following the power rule.

This results in the integrated expression of \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\).

Antiderivatives

Antiderivatives, sometimes known as indefinite integrals, are functions that reverse the process of differentiation. They help us determine the original function from its derivative.
Finding the antiderivative involves figuring out what function, when differentiated, yields the given expression.

• It's essential to remember that each function has an infinite number of antiderivatives, distinguished by the constant of integration \(C\).
• The general form for this exercise is \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\).

In practical terms, considering our integral \(\int (1-x^2-3x^5) \, dx\), each term is integrated separately to find their antiderivatives:

• \(\int 1 \, dx\) yields \(x\).
• \(\int -x^2 \, dx\) gives \(-\frac{x^3}{3}\).
• \(\int -3x^5 \, dx\) results in \(-\frac{x^6}{2}\).

Combining these gives a succinct representation of the original function, encapsulated by the unknown constant \(C\).

Differentiation Verification

Verification through differentiation is a crucial step in ensuring the correctness of an antiderivative solution.
In this method, we differentiate the antiderivative to see if we accurately arrive back at the original function.

• For \(x - \frac{x^3}{3} - \frac{x^6}{2} + C\), its derivative should match the original expression \(1-x^2-3x^5\).
• Differentiating \(x\) gives \(1\).
• The derivative of \(-\frac{x^3}{3}\) results in \(-x^2\).
• Finally, differentiating \(-\frac{x^6}{2}\) provides \(-3x^5\).

Combining these results confirms that:

• \(1 - x^2 - 3x^5\) is indeed the derivative of the proposed antiderivative.
• This process assures us that our integration using the power rule was applied correctly.

Differentiation verification is a powerful tool that acts as a mathematical checksum for our solution, validating each term's integration step.