Problem 44 - 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: Understand the Problem

The task is to find the indefinite integral of the polynomial function \(x^2 + 4x - 5\). This requires finding the antiderivative of each term separately.

Step 2: Apply the Power Rule to Each Term

The power rule for integration states that \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \(n eq -1\). Apply this rule to each term of the polynomial.

Step 3: Integrate the First Term

For the first term \(x^2\), the integral is \(\frac{x^{2+1}}{2+1} = \frac{x^3}{3}\).

Step 4: Integrate the Second Term

For the second term \(4x\), the integral is \(4 \cdot \frac{x^{1+1}}{1+1} = 4 \cdot \frac{x^2}{2} = 2x^2\).

Step 5: Integrate the Third Term

For the third term \(-5\), which is a constant, the integral is \(-5x\), since \( \int a \cdot dx = ax + C\), where \(a\) is a constant.

Step 6: Combine the Integrals

Combine all the results to get the final expression: \(\frac{x^3}{3} + 2x^2 - 5x + C\), where \(C\) is the constant of integration.

Key Concepts

Polynomial Integration

Integration is a process where we find the antiderivative or the indefinite integral of a function. When dealing with polynomials, this process involves integrating each term separately.
Polynomials are expressions consisting of variables raised to whole number powers, like the expression in our original exercise:

• \(x^2\)
• \(4x\)
• -5

Finding the indefinite integral of a polynomial means we need to find the integral of each term independently. When solving, make sure to treat each variable term according to its power and treat constant terms using the rules of constant integration. This step-by-step handling ensures accuracy and helps in understanding how each part of the polynomial contributes to the integral.

Power Rule for Integration

The power rule for integration is a key tool when integrating polynomial functions. It provides a formula that simplifies finding integrals of terms with powers.
The power rule states:

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

Here's how it works:- Take a term like \(x^2\). According to the power rule, the integral is computed by adding 1 to the exponent \(n\), resulting in \(x^{3}\). Then, divide by this new exponent, \(\frac{x^{3}}{3}\).- For terms like \(4x\), where \(n=1\), add 1 to get \(x^2\) and then divide by 2, multiply by the constant 4, to get \(2x^2\).This method ensures each powered term is integrated correctly and helps in easily identifying the structure of the antiderivative.

Constant of Integration

The constant of integration, represented as \(C\), is a crucial aspect of finding indefinite integrals. In the world of antiderivatives, any constant added to a function’s derivative won’t change the derivative itself.
That's why indefinite integrals always include "+ C."- This constant accounts for any fixed value potentially present in the original function before differentiation that would otherwise be "lost."- When given a solution like \(\frac{x^3}{3} + 2x^2 - 5x + C\), the \(C\) ensures completeness.Remember:

• Every indefinite integral has a family of functions it represents, all differing by some constant \(C\).
• The concept of \(C\) is vital because it acknowledges that not all functions start from the same point on a graph.

Understanding this idea is key to grasping why indefinite integrals are fundamental in calculus.