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.

Displacement is the integral of velocity with respect to time. $\Delta s = \int_{t_1}^{t_2} v(t) \, dt$

Substituting the given equation and limits: $\Delta s = \int_{1}^{3} (3t^2 - 2t + 1) \, dt$

Integrating term by term using the power rule $\int t^n \, dt = \frac{t^{n+1}}{n+1}$: $\Delta s = \left[ t^3 - t^2 + t \right]_{1}^{3}$

Now, substitute the upper and lower limits: At $t = 3$: $3^3 - 3^2 + 3 = 27 - 9 + 3 = 21$

At $t = 1$: $1^3 - 1^2 + 1 = 1$

Subtracting the lower limit value from the upper limit value: $\Delta s = 21 - 1 = 20\text{ m}$

• Option A is correct because the definite integration yields exactly $20\text{ m}$.
• Option B, C, and D are incorrect due to common mathematical errors, such as forgetting to subtract the lower limit ($21$ m) or incorrect differentiation instead of integration ($v'(t) = 6t - 2$).

We can solve this indefinite integral by integrating each term separately: $\int (\sin x + \sec^2 x) \, dx = \int \sin x \, dx + \int \sec^2 x \, dx$

Using standard trigonometric integration formulas:

1. $\int \sin x \, dx = -\cos x + C_1$ (since the derivative of $-\cos x$ is $\sin x$)
2. $\int \sec^2 x \, dx = \tan x + C_2$ (since the derivative of $\tan x$ is $\sec^2 x$)

Combining these results and introducing a single constant of integration $C$: $\int (\sin x + \sec^2 x) \, dx = -\cos x + \tan x + C$

• Option A is correct.
• Option B is incorrect because $\int \sin x \, dx = -\cos x$, not $\cos x$ (confusing integration with differentiation).
• Option C is incorrect because the integral of $\sec^2 x$ is $+\tan x$, not $-\tan x$.
• Option D has incorrect signs for both terms.

To solve this definite integral, we can use the substitution method. Let $u = \sin\theta$. Then, the differential is: $du = \cos\theta \, d\theta$

Now, we change the limits of integration according to our substitution:

• When $\theta = 0$, $u = \sin(0) = 0$
• When $\theta = \pi/2$, $u = \sin(\pi/2) = 1$

Substituting these values into the integral: $\int_{0}^{1} u \, du = \left[ \frac{u^2}{2} \right]_{0}^{1}$ $\left[ \frac{1^2}{2} - \frac{0^2}{2} \right] = \frac{1}{2}$

Alternative Method: Using double-angle identity $\sin\theta \cos\theta = \frac{1}{2}\sin(2\theta)$: $\int_{0}^{\pi/2} \frac{1}{2}\sin(2\theta) \, d\theta = \frac{1}{2} \left[ -\frac{\cos(2\theta)}{2} \right]_{0}^{\pi/2} = -\frac{1}{4} [\cos(\pi) - \cos(0)] = -\frac{1}{4} [-1 - 1] = \frac{1}{2}$

• Option A is correct.
• Option B is incorrect as it neglects the factor of $1/2$ from the integration of $u$.
• Option C is incorrect and typically arises from a sign error in evaluating cosine values.
• Option D is incorrect and comes from double-counting the division factor in the trigonometric identity method.