📝 Antiderivatives definition and examples (25 MCQs)

📖 Explanation: The antiderivative is found by reversing the power rule: $\int 3x^2 \, dx = x^3$, $\int -4x \, dx = -2x^2$, and $\int 5 \, dx = 5x$. Remember to always add the constant of integration $C$. Option B is correct because it includes $C$. Option A is incorrect because it omits the constant; Option C is the derivative, not the antiderivative; Option D incorrectly places the constant as a coefficient of $x$.

📖 Explanation: This tests the core theorem of antiderivatives: If two functions have the same derivative on an interval, they differ by a constant. This is because the derivative of their difference is zero. Option A is incorrect because they could be shifted vertically (e.g., $x^2$ and $x^2 + 5$). Option C and D are incorrect as they represent specific, non-general relationships. This Medium is fundamental to integration.

📖 Explanation: This applies the concept of antiderivatives to a real-world physics context. Since s'(t) = v(t), $s(t)$ is the antiderivative of $v(t)$: $\int 12t^2 dt = 4t^3 + C$. Using the initial condition $s(0) = 10$, we find $C = 10$. Option B is correct. Option A ignores the initial position; C and D have incorrect coefficients due to improper Easy of the power rule.

📖 Explanation: This question targets a common mistake: confusing the integral of $x^{-1}$ with other powers. The derivative of $\ln(x^2)$ is $2/x$, proving A is correct. The formula $\int x^n dx = x^{n+1}/(n+1)$ applies for $n \neq -1$; here $n = -2$, giving $-1/x$. The domain issue in B is also a valid concern. Option D is correct because all statements A, B, and C are valid criticisms of the student's work.

📖 Explanation: The general antiderivative is $\tan x + C$. We use the point $(\pi/4, 2)$ to find $C$: $\tan(\pi/4) + C = 2 \Rightarrow 1 + C = 2 \Rightarrow C = 1$. Therefore, the specific function is $\tan x + 1$. Option A is correct. Options C and D are common errors if one misapplies the tangent value or signs. Option B is the general form, not the specific one satisfying the condition.

📖 Explanation: Given f'(x) = 4, the antiderivative is $f(x) = 4x + C$. Using the initial condition $f(0) = -3$, we get $4(0) + C = -3 \Rightarrow C = -3$. So $f(x) = 4x - 3$. Option B is correct. Option A ignores the initial condition. Option C is the derivative itself. Option D is the general antiderivative, but not the specific function requested.

📖 Explanation: The derivative of $\sin x + C$ is $\cos x$. Options A, B, and D are all of the form $\sin x + C$, so they are antiderivatives. Option C, $-\cos x$, has a derivative of $\sin x$, not $\cos x$. Therefore, it is not an antiderivative of $\cos x$. This question tests recognition of the standard derivative $\frac{d}{dx}[\sin x] = \cos x$ and the 'plus C' principle.

📖 Explanation: This is a Easy problem. Integrate f''(x) to find f'(x) = 3x^2 + C_1. Using f'(0) = 1, we get $C_1 = 1$, so f'(x) = 3x^2 + 1. Integrate again: $f(x) = x^3 + x + C_2$. Using $f(0) = 2$, we find $C_2 = 2$. The final function is $f(x) = x^3 + x + 2$. Option A is correct. Option B is the first derivative. Option C lacks the linear term and constant. Option D is the general form before applying the second condition.

📖 Explanation: If $f(x)$ is a line with slope 3, then $f(x) = 3x + b$. Its antiderivative is $F(x) = \frac{3}{2}x^2 + bx + C$, which is a quadratic function (a parabola). Since the coefficient of $x^2$ is positive (3/2 > 0), it opens upwards. Option A is correct. Option B would be the graph of $f(x)$, not its antiderivative. Option C would correspond to a negative slope for $f(x)$. Option D would result from a quadratic $f(x)$.

📖 Explanation: The family of antiderivatives for $f(x) = 2x$ is $F(x) = x^2 + C$. To find the specific curve, we substitute the point (1, 4): $1^2 + C = 4 \Rightarrow C = 3$. So the equation is $y = x^2 + 3$. Option B is correct. Option A is a common arithmetic error. Option C is the general family, and D represents a misunderstanding of substitution.

📖 Explanation: The antiderivatives of $3x^2$ are $x^3 + C$. Any function of this form is an integral curve. Options A, B, and D follow this form (with C = 2, -1/2, and ln2, respectively). Option C, $x^3 + Cx$, has a derivative of $3x^2 + C$, not $3x^2$. Therefore, it is not an integral curve. This question tests the understanding that the constant of integration must be a constant, not a coefficient or a function of x.

📖 Explanation: The power rule for integration is $\int x^n dx = x^{n+1}/(n+1) + C$. Applying this to $\int 5x^4 dx$, we get $5 \cdot x^5/5 + C = x^5 + C$. The constant factor '5' is canceled by the division by the new exponent. The first student is correct. The second student made the common error of not dividing by the new exponent. Option C is incomplete as it omits the constant of integration.

📖 Explanation: A slope field shows $dy/dx$ at different points. If the slopes are constant, it means $dy/dx = m$, where $m$ is a constant. The solution to this differential equation is $y = mx + C$, which is a linear function. This connects the graphical representation of derivatives to the algebraic form of the antiderivative. Option B is correct. A quadratic function would have a linear slope field (slopes changing linearly), and an exponential would have a slope field proportional to y.

📖 Explanation: Rewrite $f(x) = x^{-1/2}$. The general antiderivative is $F(x) = \frac{x^{1/2}}{1/2} + C = 2\sqrt{x} + C$. Use the condition $F(4) = 5$: $2\sqrt{4} + C = 5 \Rightarrow 4 + C = 5 \Rightarrow C = 1$. Therefore, $F(x) = 2\sqrt{x} + 1$. Option A is correct. Option B incorrectly uses C=5. Option C is the derivative, and D uses C=4, which is a common arithmetic mistake.

📖 Explanation: This question requires connecting the concept of antiderivatives with a graphical interpretation of integration. Since s'(t) = v(t), $s(t)$ is the antiderivative of $v(t)$. By the Fundamental Theorem of Calculus, the integral of $v(t)$ from 0 to a gives the net displacement, which is the area under the velocity curve. Option B is correct. Option A describes acceleration. Option C is the opposite operation. Option D is a common conceptual error.

📖 Explanation: This tests knowledge of derivative rules for trigonometric functions. We know $\frac{d}{dx}[\sec x] = \sec x \tan x$. Therefore, the antiderivative of $\sec x \tan x$ is $\sec x + C$. Option C is correct. Option A is the derivative of $\sec x$, not the antiderivative. Option B is the integral of $\sec x$. Option D is the integral of $\tan x$. This question helps distinguish between common trigonometric integrals and derivatives.

📖 Explanation: The area under a curve $y = h(x)$ from 0 to 3 is $\int_0^3 h(x) dx$. For $f(x)=2x$, the area is $[x^2]_0^3 = 9$. For $g(x)=2x+5$, the area is $[x^2 + 5x]_0^3 = 9 + 15 = 24$. The difference is 15. This 15 represents the area of a rectangle of height 5 and width 3, which is the contribution of the '+5' in $g(x)$. Option B is correct. Option C incorrectly calculates the difference based only on the constant term.

📖 Explanation: This is an advanced conceptual question. If $f(x)$ is odd, then $f(-x) = -f(x)$. The derivative of $F(x) + F(-x)$ is $f(x) - f(x) = 0$, so $F(x) + F(-x) = C$, which implies $F(-x) = C - F(x)$. This means $F(x)$ is an even function plus a constant (if C=0, it is even). Option C is correct. This demonstrates a deeper connection between the symmetry of a function and its antiderivative. Option A ignores the possibility of the constant C.

📖 Explanation: Differentiating $\sqrt{x^2+1}$ involves the chain rule. Let $u = x^2+1$, $du/dx = 2x$. Then $d/dx [\sqrt{u}] = \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx} = \frac{1}{2\sqrt{x^2+1}} \cdot 2x = \frac{x}{\sqrt{x^2+1}}$. The result matches the original function, confirming the student's work is correct. Option A is correct. Option B shows a mistake in applying the chain rule. Option C and D are incorrect because the derivative of a constant is zero.

📖 Explanation: The condition 'passes through the origin' gives $f(0)=0$. The general antiderivative is $F(x) = \frac{x^3}{3} - x + C$. Using $f(0)=0$, we get $C=0$. The slope condition f'(0)=0 is $0^2 - 1 = -1 \neq 0$, which is a contradiction; it indicates a potential error in the problem statement or tests the student's ability to recognize inconsistencies. However, based on the consistent data, the function is $\frac{x^3}{3} - x$. Option A is correct. Option B would satisfy $f(0)=1$, not 0. Option C is the derivative. Option D is the general form.

📖 Explanation: Acceleration $a(t) = 50$. Velocity is the antiderivative of acceleration: $v(t) = \int 50 dt = 50t + C$. Using the initial condition $v(0)=100$, we find $C=100$. Therefore, $v(t)=50t+100$. Option A is correct. Option B is the position function (antiderivative of velocity). Option C is just the acceleration. Option D is the general form without applying the initial condition. This shows a direct Easy of the motion formulas.

📖 Explanation: This is a classic example where two different methods yield answers that differ by a constant. The derivative of $-\frac{1}{2}\cos^2 x$ is $-\frac{1}{2} \cdot 2 \cos x \cdot (-\sin x) = \sin x \cos x$. The derivative of $\frac{1}{2}\sin^2 x$ is $\frac{1}{2} \cdot 2 \sin x \cos x = \sin x \cos x$. Both are valid antiderivatives. They differ by a constant: $\frac{1}{2}\sin^2 x - (-\frac{1}{2}\cos^2 x) = \frac{1}{2}(\sin^2 x + \cos^2 x) = \frac{1}{2}$. Option C is correct. This highlights that the '+C' accounts for such differences.

📖 Explanation: This question involves the Fundamental Theorem of Calculus, which explicitly links antiderivatives and integrals. The theorem states that $\frac{d}{dx} \int_a^x f(t) dt = f(x)$. Here, $f(t) = \sin(t^2)$. So the derivative of the integral is simply $\sin(x^2)$. Option A is correct. Option B, C are results of applying the chain rule incorrectly to the integrand, which is a common mistake. Option D is the original function.

📖 Explanation: The general solution to $dy/dx = x$ is $y = \frac{x^2}{2} + C$. Option A is $\frac{x^2}{2} + 1$, which is a specific solution with C=1. Option C is $\frac{x^2}{2}$, which is another specific solution with C=0. Both are members of the family of solution curves. Option B is not correct because its derivative is 1, not x. This question checks understanding that all vertical translations of the parabola are solutions.

📖 Explanation: This question tests the Medium of the u-substitution method, which is the reverse of the chain rule. The correct formula is \int f(g(x)) g'(x) dx = \int f(u) du = F(u) + C = F(g(x)) + C. The key is that g'(x)dx must be part of the integrand to be substituted with $du$. Option A is correct. Option B ignores the necessary g'(x) factor. Option C misplaces the derivative. Option D incorrectly changes the variable of the antiderivative back to x without substitution.