Problem 61 - Applied Calculus

Question

Find the indefinite integrals. $$\int\left(x^{2}+4 x-5\right) d x$$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{x^3}{3} + 2x^2 - 5x + C \).

Step 1: Break Down the Integral

The function you need to integrate is a polynomial: x² + 4x - 5. To find the indefinite integral, you will integrate each term separately.

Step 2: Integrate the First Term

The first term is x². Use the power rule for integration, which states:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] Applying this to x² gives:\[ \frac{x^{2+1}}{2+1} = \frac{x^3}{3} \].

Step 3: Integrate the Second Term

The second term is 4x. Again, apply the power rule for integration:\[ \int 4x \, dx = 4 \cdot \frac{x^{1+1}}{1+1} = 2x^2 \] Multiply the result by 4.

Step 4: Integrate the Third Term

The third term is -5. The integral of a constant a is:\[ \int a \, dx = ax \]Applying this to -5 gives:\[ -5x \].

Step 5: Combine the Integrals

Sum up all the individual integrals you've calculated:\[ \frac{x^3}{3} + 2x^2 - 5x \] Don't forget to add the constant of integration, denoted by C, to account for any constant term that could have been differentiated to zero.

Step 6: Final Expression

The indefinite integral of the polynomial function x² + 4x - 5 is:\[ \int (x^2 + 4x - 5) \, dx = \frac{x^3}{3} + 2x^2 - 5x + C \].

Key Concepts

Polynomial Integration

Polynomial integration involves finding the antiderivative, also known as the indefinite integral, of a polynomial function. Polynomials are algebraic expressions that consist of variables and coefficients, and look like sums of various powers of the same variable. In the given exercise, we are tasked with integrating the polynomial \(x^2 + 4x - 5\).
How do we do that? By tackling each term of the polynomial individually. The process involves applying integration rules, especially focusing on the power rule, to each term.
By breaking the polynomial into its separate components \(x^2\), \(4x\), and \(-5\), we can integrate each term one by one. Once the integration of each term is completed, the results are combined to form the indefinite integral of the original polynomial function.

Power Rule for Integration

The power rule is one of the simplest and most frequently used rules for integration. It provides a straightforward way to integrate terms where the variable is raised to a power. The rule is expressed as:

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

Where \(n\) is any real number except -1, and \(C\) is the constant of integration.
In the original exercise, the power rule is applied first to \(x^2\), resulting in \(\frac{x^3}{3}\). It is also applied to \(4x\) by considering \(4\) as a constant multiplier of \(x^1\), yielding \(2x^2\) after integration. The use of the power rule simplifies the integration of polynomial terms and provides a methodical way to handle any polynomial function, making the process less intimidating for students.

Constant of Integration

The constant of integration \(C\) is a crucial part of finding indefinite integrals. When we integrate, we are essentially finding a family of functions whose derivatives give us the original function. Because differentiation of a constant results in zero, any constant could have been differentiated away in the original function.
This means there are infinitely many antiderivatives for a function, each differing by a constant. Therefore, we represent this "unknown" constant using \(C\), when stating the indefinite integral.
In our example, after finding the indefinite integral of \(x^2 + 4x - 5\), the expression \( \frac{x^3}{3} + 2x^2 - 5x + C \) includes \(C\) to acknowledge all possible antiderivatives of the polynomial, providing a comprehensive solution to the indefinite integral.