Question

If \(\int f^'(x)\cdot e^{x^2}\,dx = (x-1)\cdot e^{x^2}+k\), where \(k\) is constant of integration, then \(f(x) = \ldots\)

• A. \(2x^3-\frac{x^2}{2}+x+c\), where \(c\) is constant of integration.
• B. \(\frac{x^3}{2}+3x^2+4x+c\), where \(c\) is constant of integration.
• C. \(x^3+4x^2+6x+c\), where \(c\) is constant of integration.
• D. \(\frac{2x^3}{3}-x^2+x+c\), where \(c\) is constant of integration.

Hint:

Differentiate both sides to get f'(x)e^(x^2), then integrate f'(x).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
If an integral equals a given function plus a constant, differentiating the equation returns the integrand.

Step 2: Key Formula or Approach:
If \(\int g(x)\,dx = G(x) + k\), then \(g(x) = G'(x)\). Here \(G(x) = (x - 1)e^{x^2}\).

Step 3: Detailed Explanation:
Differentiate \(G\) using the product rule:
\[ G'(x) = e^{x^2} + (x-1)\cdot 2x\,e^{x^2} = e^{x^2}\left(1 + 2x^2 - 2x\right) \]
The integrand is \(f'(x)e^{x^2}\), so
\[ f'(x) = 2x^2 - 2x + 1 \]
Integrate to get \(f(x)\):
\[ f(x) = \frac{2x^3}{3} - x^2 + x + c \]
This is exactly option (D). Option (A) has the wrong coefficients \(2x^3\) and \(\tfrac{x^2}{2}\). Options (B) and (C) have positive \(x^2\) terms that do not come from \(-2x\).

Final Answer:
\(f(x) = \dfrac{2x^3}{3} - x^2 + x + c\), option (D). \[ \boxed{\frac{2x^3}{3}-x^2+x+c \text{ (D)}} \]