Question

If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

For integrals of the form \( \int e^x g(x) \, dx \), use the property \( \int e^x g(x) \, dx = e^x G(x) + C \), where \( G(x) \) is the integral of \( g(x) \).

Answer 1

The Correct Option is D

Step 1: Understanding the Given Integral We are given: \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C \] Using the standard integral formula: \[ \int e^x g(x) \, dx = e^x G(x) + C, \] where \( G(x) \) is the integral of \( g(x) \), we identify: \[ f(x) = \int (x^3 + x^2 - x + 4) \, dx \] 

 Step 2: Compute \( f(x) \) \[ f(x) = \int (x^3 + x^2 - x + 4) \, dx \] Integrating each term: \[ \int x^3 \, dx = \frac{x^4}{4}, \quad \int x^2 \, dx = \frac{x^3}{3}, \quad \int (-x) \, dx = -\frac{x^2}{2}, \quad \int 4 \, dx = 4x \] Thus, \[ f(x) = \frac{x^4}{4} + \frac{x^3}{3} - \frac{x^2}{2} + 4x \]  

Step 3: Evaluate \( f(1) \) \[ f(1) = \frac{(1)^4}{4} + \frac{(1)^3}{3} - \frac{(1)^2}{2} + 4(1) \] \[ = \frac{1}{4} + \frac{1}{3} - \frac{1}{2} + 4 \] Taking LCM (12): \[ = \frac{3}{12} + \frac{4}{12} - \frac{6}{12} + 4 \] \[ = \frac{1}{12} + 4 = 3 \] Thus, \( f(1) = 3 \).