Question

If \( \int_{0}^{1} (5x^2 - 3x + k) \, dx = 0 \), then \( k = \)

• A. \( \dfrac{1}{3} \)
• B. \( \dfrac{1}{6} \)
• C. \( -\dfrac{1}{3} \)
• D. \( -\dfrac{1}{6} \)

Hint:

When a definite integral equals zero, always simplify carefully before solving for constants.

Answer 1

The Correct Option is D

Step 1: Integrate the expression.
\[ \int_{0}^{1} (5x^2 - 3x + k) \, dx \] \[ = \left[ \frac{5x^3}{3} - \frac{3x^2}{2} + kx \right]_0^1 \] Step 2: Apply the limits.
\[ \frac{5}{3} - \frac{3}{2} + k = 0 \] Step 3: Simplify.
\[ \frac{10 - 9}{6} + k = 0 \] \[ \frac{1}{6} + k = 0 \] Step 4: Solve for \( k \).
\[ k = -\frac{1}{6} \] Step 5: Conclusion.
The value of \( k \) is \( -\dfrac{1}{6} \).