CA Foundation · Quantitative Aptitude · Differential and Integral Calculus
Evaluate ∫(8x³ − 6x + 5) dx.
Using the power rule for indefinite integrals, ∫8x³ dx = 2x⁴, ∫(−6x) dx = −3x², and ∫5 dx = 5x. The complete antiderivative is 2x⁴ − 3x² + 5x + C, where C is an arbitrary constant.
- A2x⁴ − 3x² + 5x + CCorrect
- B2x⁴ − 6x² + 5x + C
- C4x⁴ − 3x² + 5x + C
- D2x⁴ − 3x + C
Explanation
Using the power rule for integration: ∫8x³ dx = 8·x⁴/4 = 2x⁴, ∫(−6x) dx = −6·x²/2 = −3x², and ∫5 dx = 5x. Therefore, the antiderivative is 2x⁴ − 3x² + 5x + C. Option 1 incorrectly integrates −6x as −6x², and option 3 uses wrong coefficient for x⁴.
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