Question

Evaluate: $\int x \cdot e^x \, dx$

• A. \( x e^x + C \)
• B. \( e^x(x - 1) + C \)
• C. \( e^x(x + 1) + C \)
• D. \( x^2 e^x + C \)

Hint:

For integrals of the form \(\int P(x) e^x \, dx\) where \(P(x)\) is a polynomial, you can use tabular integration (DI method): alternatingly write the polynomial and its derivatives alongside the integrals of \(e^x\), then multiply across diagonals. \( (+)[x \cdot e^x] - (-)[1 \cdot e^x] = e^x(x-1) + C \).

Answer 1

The Correct Option is B

Concept: To integrate the product of an algebraic function and an exponential function, we apply the Integration by Parts method. The formula is stated as: \[ \int u \cdot v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \int v \, dx \right) dx \] We use the ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential) to select the first function (\(u\)). In this case:

• First function, \(u = x\) (Algebraic)
• Second function, \(v = e^x\) (Exponential)

Step 1: Applying the Integration by Parts formula.
Differentiating \(u\) and integrating \(v\): \[ \frac{du}{dx} = 1 \quad \text{and} \quad \int e^x \, dx = e^x \] Substituting these components into the integration by parts formula: \[ \int x \cdot e^x \, dx = x \cdot \left(e^x\right) - \int \left( 1 \cdot e^x \right) dx \]

Step 2: Evaluating the remaining integral and simplifying.
\[ \int x \cdot e^x \, dx = x e^x - \int e^x \, dx \] \[ \int x \cdot e^x \, dx = x e^x - e^x + C \] Factoring out the common exponential term \(e^x\): \[ \int x \cdot e^x \, dx = e^x(x - 1) + C \]