Question:
If \( \int_0^1 (6x^2 - 4x + k) \, dx = 0 \), then the value of \( k \) is :
If \( \int_0^1 (6x^2 - 4x + k) \, dx = 0 \), then the value of \( k \) is :
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For polynomial integration, integrate term by term. If the lower limit is zero, terms containing \( x \) will vanish, making the calculation dependent only on the upper limit.
Updated On: Sep 10, 2026
- \( 1 \)
- \( 0 \)
- \( 2 \)
- \( -1 \)
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The Correct Option is B
Solution and Explanation
Concept:
• Power Rule of Integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \).
• Fundamental Theorem of Calculus: \( \int_a^b f(x) \, dx = F(b) - F(a) \), where \( F \) is the antiderivative.
Step 1: Find the antiderivative of the given function
The given integral is \( \int_0^1 (6x^2 - 4x + k) \, dx \).
Apply the sum rule and power rule for integration:
\[ \int (6x^2 - 4x + k) \, dx = 6\left(\frac{x^3}{3}\right) - 4\left(\frac{x^2}{2}\right) + kx \]
Simplify the expression:
\[ = 2x^3 - 2x^2 + kx \]
Step 2: Apply the limits of integration
Substitute the upper limit \( 1 \) and the lower limit \( 0 \) into the antiderivative:
\[ [2x^3 - 2x^2 + kx]_0^1 = (2(1)^3 - 2(1)^2 + k(1)) - (2(0)^3 - 2(0)^2 + k(0)) \]
Simplify the resulting expression:
\[ = (2 - 2 + k) - 0 = k \]
Step 3: Equate to zero and solve for \( k \)
The problem states that the definite integral is equal to zero.
\[ k = 0 \]
• Power Rule of Integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \).
• Fundamental Theorem of Calculus: \( \int_a^b f(x) \, dx = F(b) - F(a) \), where \( F \) is the antiderivative.
Step 1: Find the antiderivative of the given function
The given integral is \( \int_0^1 (6x^2 - 4x + k) \, dx \).
Apply the sum rule and power rule for integration:
\[ \int (6x^2 - 4x + k) \, dx = 6\left(\frac{x^3}{3}\right) - 4\left(\frac{x^2}{2}\right) + kx \]
Simplify the expression:
\[ = 2x^3 - 2x^2 + kx \]
Step 2: Apply the limits of integration
Substitute the upper limit \( 1 \) and the lower limit \( 0 \) into the antiderivative:
\[ [2x^3 - 2x^2 + kx]_0^1 = (2(1)^3 - 2(1)^2 + k(1)) - (2(0)^3 - 2(0)^2 + k(0)) \]
Simplify the resulting expression:
\[ = (2 - 2 + k) - 0 = k \]
Step 3: Equate to zero and solve for \( k \)
The problem states that the definite integral is equal to zero.
\[ k = 0 \]
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