Problem 22 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. 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: Identify Each Term to Integrate

First, we identify the individual terms in the integrand: 1, \(-x^2\), and \(-3x^5\). Our goal is to find the antiderivative of each term separately.

Step 2: Apply Basic Integration Rules

We use the basic power rule for integrals: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). This applies to each term:\[ \int 1 \, dx = x + C_1 \] \[ \int (-x^2) \, dx = -\frac{x^{3}}{3} + C_2 \] \[ \int (-3x^5) \, dx = -\frac{3x^{6}}{6} + C_3 = -\frac{x^6}{2} + C_3 \]

Step 3: Combine Results and Simplify

Combine the antiderivatives of each term into a single expression including a constant of integration. Simplify the expression if necessary.\[ \int (1 - x^2 - 3x^5) \, dx = x - \frac{x^3}{3} - \frac{x^6}{2} + C \] where \( C = C_1 + C_2 + C_3 \) is an arbitrary constant.

Step 4: Check by Differentiation

Differentiate the result to verify it's correct. The derivative should match the original integrand.\[ \frac{d}{dx} \left( x - \frac{x^3}{3} - \frac{x^6}{2} + C \right) = 1 - x^2 - 3x^5 \] This matches with the original integrand, confirming our antiderivative is correct.

Key Concepts

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus, representing the family of all antiderivatives of a given function. When you see the notation \( \int f(x) \, dx \), it asks for an antiderivative of the function \( f(x) \). An indefinite integral does not have limits of integration, meaning that it represents a general form involving the variable plus a constant of integration, \( C \).

• The antiderivative is a way to reverse differentiation.
• Indefinite integrals always include the constant \( C \) because the derivative of any constant is zero.
• The result should be checked by differentiating the function we found; the derivative must match the original function.

For example, when integrating a function like \( \int (1 - x^2 - 3x^5) \, dx \), we find the antiderivative of each term and then combine these terms to form a general solution. Always remember to add the constant of integration \( C \) in your final expression.

Integration Techniques

Integration techniques such as substitution and parts can help solve more complex integrals. However, for simpler polynomials, like \( 1-x^{2}-3 x^{5} \), we often use basic rules and principles.When we approach an integral, we typically break it into smaller parts if necessary, then find the integrals of these parts individually. This is called term-by-term integration.

• First, identify the individual terms within the expression that require integration.
• Apply basic integral rules to each term.
• Combine the results and simplify wherever possible.

As seen in the original solution, integration was performed separately for terms like \( 1 \), \(-x^2\), and \(-3x^5\). By doing this systematically, we ensure that the integration is not only easier to manage but also correct when checked by differentiation.

Power Rule for Integrals

The power rule is one of the simplest yet most crucial tools in calculus for finding indefinite integrals. It states that for any real number \( n eq -1 \), the integral of \( x^n \) is formed by adding one to the exponent and dividing by the new exponent: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]
This rule is particularly useful with polynomial functions. Here is how it applies to the integral \( \int (1 - x^2 - 3x^5) \, dx \):

• For the constant term \( 1 \), the integral is simply \( x \), since \( \int 1 \, dx = x + C \).
• For \(-x^2\), apply the rule: \( \int -x^2 \, dx = -\frac{x^{3}}{3} + C \).
• For \(-3x^5\), use the rule: \( \int -3x^5 \, dx = -\frac{3x^{6}}{6} + C = -\frac{x^6}{2} + C \).

By calculating the integral of each term separately and then combining them, we arrive at the correct and simplified form of the antiderivative. This shows how powerful and straightforward the power rule truly is when dealing with polynomials.