Problem 14 - Applied Calculus

Question

Use integration by parts to find each integral. \(\int(x-1) e^{x} d x \quad\)

Step-by-Step Solution

Answer

The integral is \(xe^x - 2e^x + C\).

Step 1: Identify Parts for Integration by Parts

The formula for integration by parts is \(\int u \, dv = uv - \int v \, du\). We need to choose \(u\) and \(dv\). For this integral, let \(u = x-1\) and \(dv = e^{x} \, dx\).

Step 2: Differentiate and Integrate Parts

Differentiate \(u\) and integrate \(dv\). \(du = d(x-1) = dx\). Now integrate \(dv\) to get \(v = \int e^x \, dx = e^x\).

Step 3: Apply the Integration by Parts Formula

Substitute \(u\), \(v\), \(du\), and \(dv\) back into the integration by parts formula: \(\int (x-1) e^{x} \, dx = (x-1) e^x - \int e^x \, dx\).

Step 4: Evaluate the Remaining Integral

Evaluate the integral \(\int e^x \, dx\) to get \(e^x\). Substitute this back into the equation to get \((x-1)e^x - e^x\).

Step 5: Simplify the Expression

Simplify the expression \((x-1)e^x - e^x\) to \(xe^x - 2e^x\).

Step 6: Add the Constant of Integration

Since this is an indefinite integral, remember to add the constant of integration. The final result is \(xe^x - 2e^x + C\).

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, represents a family of functions whose derivatives give back the original function. The process of finding an indefinite integral is called integration. Unlike definite integrals, which find a numerical value for the area under a curve, indefinite integrals result in a general formula that includes a constant of integration, denoted by "C."

The constant of integration is essential because when you integrate a function, you're essentially determining what functions could have led to your derivative — and there are infinitely many possibilities, each differing by a constant value. Calculating an indefinite integral involves reversing the process of differentiation and represents a fundamental concept in calculus and analysis.

Integration Techniques

Integration by parts is a powerful technique based on the product rule of differentiation. Its formula is \(\int u \, dv = uv - \int v \, du\). This method is particularly useful for integrals involving products of functions, especially when one component is easily differentiable, and the other is easily integrable.

Here’s a typical way to decide on the functions for \(u\) and \(dv\):

• Choose \(u\) as the function that becomes simpler when differentiated. Typical choices include polynomial expressions or inverse trigonometric functions.
• The \(dv\) component will be the rest of the integrand, which you should be able to integrate back into \(v\).

For the integral \(\int (x-1) e^{x} \, dx\), choosing \(u = x-1\) and \(dv = e^{x} \, dx\) simplifies the process. Differentiating \(u\) and integrating \(dv\) leads you smoothly through the integration.

Step by Step Solution

To solve the integral \(\int(x-1) e^{x} \, dx\) using integration by parts:

• Assign \(u = x-1\) and differentiate to get \(du = dx\).
• Let \(dv = e^{x} \, dx\) and integrate to obtain \(v = e^{x}\).
• Substitute into the integration by parts formula: \(\int (x-1) e^{x} \, dx = (x-1)e^{x} - \int e^{x} \, dx\).

Continuing from here, calculate \(\int e^{x} \, dx\) to find \(e^{x}\). Substituting back into the equation, you get \((x-1)e^{x} - e^{x}\).

Now, simplify the expression to \(xe^{x} - 2e^{x}\). Remember, since this is an indefinite integral, you must add a constant of integration at the end, resulting in \(xe^{x} - 2e^{x} + C\). This detailed, step-by-step approach clarifies the solution and emphasizes the importance of each step in the integration process.