Questions & Answers: "Intergration"

According to the power rule of integration, ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C for n ≠ -1. Applying this to each term: ∫x³ dx = x⁴/4, ∫-4x dx = -4(x²/2) = -2x², and ∫5 dx = 5x. Summing these results and adding the constant of integration C gives x⁴/4 - 2x² + 5x + C. Option 2 is incorrect because it represents the derivative of the function. Option 3 fails to divide the coefficient 4 by the new power 2. Option 4 uses the wrong power for the first term.

The standard integral formula is ∫sin(ax) dx = -(1/a)cos(ax) + C. In this case, a = 2. Therefore, ∫sin(2x) dx = -(1/2)cos(2x) + C. Option 2 is incorrect because the integral of the sine function is negative cosine. Option 3 is a common mistake where the student multiplies by the constant (as in differentiation) instead of dividing. Option 4 neglects the coefficient introduced by the substitution method (chain rule in reverse).

To solve a definite integral, first find the indefinite integral: ∫(6x² + 2) dx = 6(x³/3) + 2x = 2x³ + 2x. Next, apply the Fundamental Theorem of Calculus by substituting the upper limit (2) and subtracting the result of substituting the lower limit (1). For x=2: 2(2)³ + 2(2) = 16 + 4 = 20. For x=1: 2(1)³ + 2(1) = 2 + 2 = 4. The final value is 20 - 4 = 16. Option 2 is just the upper limit value, and options 3 and 4 result from calculation errors in integration or subtraction.

To solve this, we apply the power rule of integration: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C. For the first term, ∫ 6x² dx = 6(x³/3) = 2x³. For the second term, ∫ -4x dx = -4(x²/2) = -2x². For the third term, ∫ 3 dx = 3x. Combining these and adding the constant of integration C, we get 2x³ - 2x² + 3x + C. Option 1 is incorrect because it represents the derivative, not the integral. Option 3 has an incorrect coefficient for the first term (3 instead of 2). Option 4 has an incorrect coefficient for the second term (4 instead of 2).

The integral of sin(x) is -cos(x). To evaluate the definite integral from 0 to π/2, we compute [-cos(x)] evaluated from 0 to π/2. This equals [-cos(π/2)] - [-cos(0)]. Since cos(π/2) = 0 and cos(0) = 1, the expression becomes [0] - [-1] = 1. Option 1 is incorrect as it represents the integral of sin(x) over a full period or certain symmetric intervals. Option 2 is a sign error. Option 4 incorrectly treats the function as a constant.

Displacement is calculated by the definite integral of velocity with respect to time: s = ∫₀² (3t² + 2t) dt. First, find the antiderivative: ∫ (3t² + 2t) dt = t³ + t². Now, apply the limits from 0 to 2: [t³ + t²] evaluated from 0 to 2 = (2³ + 2²) - (0³ + 0²) = (8 + 4) - 0 = 12 meters. Option 1 (10 m) is a common calculation error where one term is missed. Options 3 and 4 are incorrect numerical results arising from improper application of the upper limit.