Problem 21 - 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. $$\int\left(2 x^{3}-5 x+7\right) d x$$

Step-by-Step Solution

Answer

The antiderivative is \(\frac{x^4}{2} - \frac{5}{2}x^2 + 7x + C\).

Step 1: Identify the integrands

The given integral is \( \int (2x^3 - 5x + 7) \, dx \). The integrand consists of three terms: \(2x^3\), \(-5x\), and \(7\). Each term can be integrated separately following the power rule.

Step 2: Apply the power rule

To integrate each term individually, use the power rule for integration. For a term like \(ax^n\), the integral is \(\frac{a}{n+1}x^{n+1}\). Apply this to each term:- The integral of \(2x^3\) is \(\frac{2}{3+1}x^{3+1} = \frac{2}{4}x^4 = \frac{x^4}{2}\).- The integral of \(-5x\) is \(\frac{-5}{1+1}x^{1+1} = \frac{-5}{2}x^2\).- The integral of \(7\) is \(7x\), since the integral of a constant \(a\) is \(ax\).

Step 3: Combine the results

Combine the integrals from Step 2: \[ \int (2x^3 - 5x + 7) \, dx = \frac{x^4}{2} - \frac{5}{2}x^2 + 7x + C \]where \(C\) is the constant of integration.

Step 4: Verify by differentiation

Differentiate the result to ensure it gives back the original integrand:- Differentiating \(\frac{x^4}{2}\) gives \(2x^3\).- Differentiating \(-\frac{5}{2}x^2\) gives \(-5x\).- Differentiating \(7x\) gives \(7\).Combining these derivatives yields \(2x^3 - 5x + 7\), which is the original integrand, confirming our solution is correct.

Key Concepts

Power Rule for Integration

The power rule for integration is an essential tool when dealing with indefinite integrals. It makes finding the antiderivative of polynomial terms quite straightforward. The power rule states that for any term in the form of \( ax^n \), its integral can be found using the formula: \[ \int ax^n \; dx = \frac{a}{n+1} x^{n+1} + C \] where \(a\) is a constant, \(n\) is the power of \(x\), and \(C\) is a constant of integration.
The power rule only applies if \(neq -1\), as trying \(n=-1\) would lead to division by zero.
This rule essentially increases the power of \(x\) by 1 and divides the coefficient by the new power.

• In the exercise, for the term \(2x^3\), the power rule applies resulting in \(\frac{2}{4}x^4\), simply written as \(\frac{x^4}{2}\).
• Similarly, \(-5x\) becomes \(\frac{-5}{2}x^2\), and the constant term \(7\) becomes \(7x\), since the integral of a constant is the constant times \(x\).

The power rule is powerful because it allows the integration of polynomials term by term with ease.

Antiderivative

An antiderivative, often called an indefinite integral, is essentially the reverse process of differentiation. When you take the derivative of a function, you're finding the rate at which the function changes. An antiderivative tells you what function had that rate of change.
Mathematically, finding an antiderivative means solving the equation: \[ F'(x) = f(x) \] where \( F(x) \) is the function whose derivative is \( f(x) \).
In simple terms, it's the process of finding a function that, when differentiated, gives the original function.

• In our exercise, we've found that the antiderivative of \(2x^3 - 5x + 7\) is \( \frac{x^4}{2} - \frac{5}{2}x^2 + 7x + C \).
• Each term in the integrand is integrated individually using rules like the power rule to find the corresponding antiderivative term.

This process of term-by-term integration simplifies the method of finding antiderivatives, especially for polynomial functions.

Constant of Integration

The constant of integration, represented by \(C\), is an integral part of any indefinite integral result. Its presence is crucial because when differentiating, constant terms disappear.
This means for any indefinite integral, there could be infinitely many antiderivatives that differ from each other by a constant value.
Therefore, adding \(C\) accounts for all those possible antiderivatives.

• Consider the integral \(\int (2x^3 - 5x + 7) \; dx = \frac{x^4}{2} - \frac{5}{2}x^2 + 7x + C\). Here, \(C\) represents any constant that, when differentiated, equals zero.
• This universality provides a set of functions rather than a single function, with one primary rule: all have the same derivative \(2x^3 - 5x + 7\).

The constant of integration allows the result to capture the full scope of possible functions that could have the given derivative, reflecting the integral’s inherent non-uniqueness.