Problem 21 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int\left(2 x^{3}-5 x+7\right) d x $$

Step-by-Step Solution

Answer

\(\frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C\) is the most general antiderivative.

Step 1: Find the Antiderivative of Each Term

To find the antiderivative of a polynomial, apply the reverse of the power rule. For a term like \(ax^n\), the antiderivative is \(\frac{a}{n+1}x^{n+1}\). Apply this to each term in the integral.

Step 2: Antiderivative of the First Term

For the first term \(2x^3\), apply the rule \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\). Thus the antiderivative is \(\frac{2}{4}x^4 = \frac{1}{2}x^4\).

Step 3: Antiderivative of the Second Term

The second term is \(-5x\), which is of the form \(ax\). The antiderivative is \(\frac{-5}{2}x^2 = \frac{-5}{2}x^2\).

Step 4: Antiderivative of the Constant Term

A constant \(c\) integrates to \(cx\). Thus, the constant 7 integrates to \(7x\).

Step 5: Combine All Terms

Combine all antiderivatives with a constant of integration \(C\). The most general antiderivative of the given function is \(\frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C\).

Step 6: Check the Antiderivative by Differentiation

Differentiate the antiderivative \(\frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C\). The derivative of \(\frac{1}{2}x^4\) is \(2x^3\), the derivative of \(-\frac{5}{2}x^2\) is \(-5x\), and the derivative of \(7x\) is 7. Combine these to get the original function \(2x^3 - 5x + 7\), confirming the solution is correct.

Key Concepts

Antiderivative

In calculus, an antiderivative of a function is a function whose derivative is the original function. It's like working backward from differentiation to find a function that, when differentiated, gives us the function we started with. Every continuous function has an infinite number of antiderivatives, differing only by a constant. This is because the derivative of a constant is zero. For example, if we know that the derivative of some function is \f(x) = 3x^2\, then one antiderivative would be \F(x) = x^3\, since \(F'(x) = 3x^2\).

• The process of finding an antiderivative is often called "integration."
• An indefinite integral is the representation of an antiderivative, expressed using the integral symbol and including a constant of integration \(C\).

Integrating allows us to determine each possible original function from its derivative. Hence, for the function \(f(x) = 2x^3 - 5x + 7\), its general antiderivative is \(F(x) = \frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C\).

Power Rule

The power rule is a basic principle for finding antiderivatives involving polynomials. This rule helps you integrate any term of the form \(ax^n\). According to the power rule for integration, the antiderivative of such a term is \(\frac{ax^{n+1}}{n+1}\), where \(n\) is not equal to -1.

• The power rule simply requires incrementing the exponent by one and dividing by the new exponent.
• For instance, the antiderivative of \2x^3\ is obtained using this rule: \frac{2}{4}x^4 = \frac{1}{2}x^4\.
• For linear terms, like \-5x\, it becomes \frac{-5}{2}x^2\.

This rule is very handy and speeds up the integration process for polynomial functions. Its simplicity makes it an essential tool in calculus!

Polynomial Integration

Polynomial integration is the process of integrating polynomial functions using basic rules, primarily relying on the power rule. Polynomial functions are made up of terms which are products of constants and variables raised to whole number powers.

• To integrate a polynomial, apply the power rule to each term individually.
• Consider the integral of \(2x^3 - 5x + 7\) as an example.
• Each term is integrated separately using rules like \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).
• The integral of the constant term, like 7, is simplified as multiplying the constant by \(x\).

In this way, the polynomial \(2x^3 - 5x + 7\) becomes \(\frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C\), by combining all the integrated terms. Polynomial integration offers a clear and methodical approach to finding antiderivatives of polynomial expressions.