QCAA Mathematical Methods Further integration

This option neglects the constant factor of 2. It represents the value of $\int_{\pi/3}^{\pi/2} \cos(x)dx$, which is $1 - \frac{\sqrt{3}}{2}$.

This result comes from missing the factor of 2 and reversing the order of subtraction (calculating lower limit minus upper limit).

This answer results from swapping the limits of integration or subtracting in the wrong order ($F(a) - F(b)$), yielding the negative of the correct answer.

This value corresponds to $\ln(6)$. This error typically occurs if you evaluate the antiderivative at the upper limit ($x=5a$) but forget to subtract the evaluation at the lower limit ($x=a$).

This value corresponds to $\ln(5)$. This is incorrect; the integration results in the natural log of the ratio of the bounds adjusted by $a$, which is $\ln(3)$, not $\ln(5)$.

This value corresponds to $\ln(4)$. The correct evaluation of the definite integral yields $\ln(3)$.

This incorrectly subtracts $10$ from $-4$ or adds the two values incorrectly. The correct operation is to subtract the integral from $0$ to $a$ from the integral from $0$ to $b$.

This is simply the given value for $\int_0^a f(x) dx$, not the integral over the interval from $a$ to $b$.

There is no mathematical justification for the integral evaluating to $0$ based on the given values.

This is the given value for $\int_0^b f(x) dx$, which represents the area over the entire interval from $0$ to $b$, not just from $a$ to $b$.

This is the value of the definite integral $\int_{-1}^{1} xe^x \, dx = \frac{2}{e} \approx 0.74$. This calculation finds the net signed area, where the region below the x-axis cancels out part of the region above. To find the total geometric area, you must integrate the absolute value of the function.

This value corresponds to $\int_{-1}^{1} e^x \, dx = e - \frac{1}{e} \approx 2.35$. This would be the area under the graph of $y=e^x$, rather than the given function $y=xe^x$.

This value approximates $|1 \cdot e^1| + |-1 \cdot e^{-1}| = e + \frac{1}{e} \approx 3.09$. This calculation sums the absolute values of the function at the endpoints of the interval, which does not represent the area under the curve.

Incorrect. This evaluates $\int_0^a 3f(x)dx$ but completely ignores the integral of the constant $2$.

Incorrect. This correctly evaluates the integral of $2$ as $2a$, but forgets to multiply the integral of $f(x)$ by the constant factor $3$.

Incorrect. This forgets to multiply the integral of $f(x)$ by $3$ and incorrectly evaluates $\int_0^a 2dx$ as $2$ instead of $2a$.

Incorrect. This correctly multiplies the integral of $f(x)$ by $3$, but incorrectly evaluates $\int_0^a 2dx$ as $2$ instead of $2a$.

Incorrect. This result likely comes from adding the evaluated bounds ($2^3 + 1^3 = 9$) instead of subtracting them, which violates the Fundamental Theorem of Calculus.

Incorrect. This value might come from evaluating the derivative $6x$ at $x=1$, rather than finding the definite integral.

Incorrect. This is a miscalculation. Remember to find the antiderivative $x^3$ and evaluate $F(2) - F(1)$.

This value represents the acceleration at $t=4$, found by differentiating the velocity function: $a(4) = v'(4) = \frac{2}{4+1} = 0.40 \text{ m/s}^2$.

This value corresponds to $\ln(5)$, which is an intermediate value in the calculation but not the final integral result.

This is the instantaneous velocity of the marble at $t=4$, calculated as $v(4) = 2\ln(5) \approx 3.22 \text{ m/s}$, rather than the total distance traveled.

This answer is incorrect. It corresponds to evaluating an antiderivative like $\frac{1}{2}\ln(2x)$ only at the upper limit $x=3$, neglecting to subtract the value at the lower limit.

This is incorrect. The evaluation of the definite integral depends on the limits $1$ and $3$, which results in $\ln(3)$, not $\ln(5)$.

This option is incorrect. It may result from an arithmetic error or misapplying logarithmic properties, as the integral yields $\ln(3) - \ln(1)$, not $\ln(4)$.