Problem 13 - Thomas Calculus

Question

Evaluate the integrals. \(\int\left(x^{2}-5 x\right) e^{x} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \((x^2 - 7x + 7)e^x + C\).

Step 1: Identify the Integration Technique

This problem involves the integration of a product of a polynomial and an exponential function. Therefore, the best method to use is Integration by Parts, which is defined as \( \int u \, dv = uv - \int v \, du \).

Step 2: Choose Functions for Integration by Parts

To apply integration by parts, we need to identify parts of the integrand as \( u \) and \( dv \). Let \( u = x^2 - 5x \) such that \( du = (2x - 5) \, dx \). Let \( dv = e^x \, dx \), thus \( v = e^x \).

Step 3: Apply Integration by Parts Formula

According to the integration by parts formula, substitute \( u \), \( du \), \( v \), and \( dv \) into \( \int u \, dv \) to get: \( \int (x^2 - 5x) e^x \, dx = (x^2 - 5x)e^x - \int e^x (2x - 5) \, dx \).

Step 4: Simplify and Reapply Integration by Parts

The integral \( \int e^x (2x - 5) \, dx \) needs further evaluation. Apply integration by parts again with \( u = 2x - 5 \), \( du = 2 \, dx \), and \( dv = e^x \, dx \), \( v = e^x \).

Step 5: Continue with Second Integration by Parts

Using the integration by parts formula again, we get: \( \int (2x - 5)e^x \, dx = (2x - 5)e^x - \int 2e^x \, dx \). Compute the remaining integral \( \int 2e^x \, dx = 2e^x \).

Step 6: Combine All Parts for the Final Solution

Substitute back the integrals into one complete statement: \[ \int (x^2 - 5x) e^x \, dx = (x^2 - 5x)e^x - [(2x - 5)e^x - 2e^x] \]. Simplify to get: \[ (x^2 - 5x - 2x + 5)e^x + 2e^x + C \].

Step 7: Simplify the Expression

Simplify the expression: \( (x^2 - 7x + 7)e^x + C \). Thus, the solution is neat and final.

Key Concepts

Exponential Function Integration

In calculus, integrating an exponential function can sometimes be straightforward, especially when it's part of a larger, complex expression. The exponential function, typically represented as \(e^x\), is unique because it retains its form even after differentiation or integration. When integrated, the function \(e^x\) leads to the integral \(\int e^x \, dx = e^x + C\), where \(C\) is the constant of integration. This property makes it easier to handle exponential functions compared to others, integrating seamlessly across a wide range of problems.

In the context of our problem, exponential integration is paired with polynomial expressions, and invokes integration by parts. It's a common scenario where understanding exponential properties aids in breaking down more intricate integrals. Recall that when integrating by parts, it’s strategic to set \(dv = e^x \, dx\) because its integral \(v = e^x\) remains manageable, thus simplifying subsequent steps.

Polynomial Integration

Polynomial integration involves finding the antiderivative of a polynomial function. Polynomials are expressions consisting of variables and coefficients, often seen in the format \(ax^n + bx^{n-1} + ... + zx^0\). To integrate a polynomial, typically each term is integrated separately, following the power rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\). This rule is straightforward but extremely powerful in solving polynomial integrals efficiently.

However, in our problem, the polynomial \(x^2 - 5x\) is multiplied by another function, leading us to use integration by parts. Here, we choose one part of the polynomial to differentiate while integrating the exponential function. For \(x^2 - 5x\), by selecting \(u = x^2 - 5x\), we find \(du = (2x - 5) \, dx\), effectively setting up the integration by parts process. It shows how knowing basic polynomial integration aids in tackling complex integrals when combined with exponential functions.

Definite and Indefinite Integrals

When solving integrals, it's important to differentiate between definite and indefinite integrals. Indefinite integrals, like the one in our example, do not have limits of integration. They show the general form of antiderivatives and always include a constant of integration, \(C\), since they represent a family of functions.

On the other hand, definite integrals calculate the area under a curve within given limits, producing a numerical result. They are typically solved by evaluating the antiderivative at the upper and lower limits and subtracting the two values.

In our solution, the integral process focuses solely on finding the indefinite integral of the function \((x^2 - 5x)e^x\). This requires breaking down the polynomial-exponential combination using integration by parts. The result, \((x^2 - 7x + 7)e^x + C\), reveals the function's antiderivative without evaluating specific bounds. Understanding these distinctions helps avoid confusion when handling various calculus problems.