Integrate the expression 6x$^2$ - 2x + 1

Analyze the question: We need the indefinite integral of 6x² − 2x + 1 with respect to x.

Concept: Use the power rule for integration term by term:

∫ axⁿ dx = a·xⁿ⁺¹/(n+1) + c, and ∫ k dx = kx + c

Solution:

1. ∫ 6x² dx = 6 · x³/3 = 2x³
2. ∫ −2x dx = −2 · x²/2 = −x²
3. ∫ 1 dx = x
4. Add the constant of integration, c.

So: ∫(6x² − 2x + 1) dx = 2x³ − x² + x + c

Check by differentiating: d/dx (2x³ − x² + x + c) = 6x² − 2x + 1 ✓

Compare with the options:

• A: differentiates to 9x² − 4x + 1, so it is wrong.
• B: 2x³ − x² + x + c, which matches our result.
• C: differentiates to 6x² − 6x, so it is wrong.
• D: differentiates to 3x² + 2x − 1, so it is wrong.

Tip: In integration, raise the power by 1 and divide by the new power. Always add "+ c" for indefinite integrals. If you're unsure, differentiate your answer to check that you get back the original expression. This takes only a few seconds and saves marks in JAMB.