Answer: Both are correct because they differ only by a constant, and both are valid antiderivatives of the integrand

Verify the antiderivative structure: Both answers have the form $-2\cos x + \tan x + C$. Student A chose $C = 7$ and Student B chose $C = -3$. Why the correct answer works: Any two antiderivatives of the same function differ by a constant. Since both expressions differentiate back to $2\sin x + \sec^2 x$, both are correct antiderivatives. Why distractors fail: Options A and B incorrectly restrict the sign of $C$. Option D incorrectly insists on the symbolic $+C$; specifying a particular constant value is equally valid.

Answer: $8\sqrt{x} - 2e^x + 7x + C$

Rewrite and integrate term by term: $4x^{-1/2}$: $\int 4x^{-1/2}\,dx = 4 \cdot \dfrac{x^{1/2}}{1/2} = 8\sqrt{x}$. Then $\int -2e^x\,dx = -2e^x$ and $\int 7\,dx = 7x$. Why the correct answer works: Combining: $8\sqrt{x} - 2e^x + 7x + C$ (Option A). Why distractors fail: Option B differentiates $4/\sqrt{x}$ instead of integrating. Option C divides by $1/2$ incorrectly (getting $2\sqrt{x}$ instead of $8\sqrt{x}$). Option D misapplies the power rule to $e^x$.

Answer: $\dfrac{5^x}{\ln 5} + C$

Recall the exponential antiderivative for base $a$: For $a > 0$, $a \neq 1$: $\int a^x\,dx = \dfrac{a^x}{\ln a} + C$. Why the correct answer works: With $a = 5$, Option B gives $\dfrac{5^x}{\ln 5} + C$. Differentiating: $\dfrac{5^x \ln 5}{\ln 5} = 5^x$. ✓ Why distractors fail: Option A is the derivative of $5^x$, not its antiderivative. Option C misapplies the polynomial power rule. Option D also misapplies the polynomial power rule.

Answer: $\dfrac{3}{5}x^{5/3} - 3x^{-1/3} + C$

Apply the power rule to each term: $\int x^{2/3}\,dx = \dfrac{x^{5/3}}{5/3} = \dfrac{3}{5}x^{5/3}$. $\int x^{-4/3}\,dx = \dfrac{x^{-1/3}}{-1/3} = -3x^{-1/3}$. Why distractors fail: Option B subtracts 1 from $-4/3$ (getting $-7/3$) instead of adding 1. Option C inverts the fraction, multiplying by $5/3$ instead of dividing. Option D gives the derivatives, not antiderivatives.

Answer: $\dfrac{2}{3}x^{3/2} + C$

Rewrite and apply the power rule: $\sqrt{x} = x^{1/2}$, so $\int x^{1/2}\,dx = \dfrac{x^{3/2}}{3/2} = \dfrac{2}{3}x^{3/2} + C$. Why the correct answer works: Option A correctly divides by $3/2$, which is equivalent to multiplying by $2/3$. Why distractors fail: Option B is the derivative of $\sqrt{x}$, not the antiderivative. Option C multiplies by $3/2$ instead of dividing. Option D divides by 3 instead of $3/2$.

Answer: The classmate is correct: differentiating $\cos x$ gives $-\sin x \neq \sin x$, while differentiating $-\cos x$ gives $\sin x$

Verify by differentiating each candidate: $\frac{d}{dx}[\cos x] = -\sin x$ (not $\sin x$). $\frac{d}{dx}[-\cos x] = \sin x$ ✓. Why the correct answer works: Option B correctly identifies that the classmate's answer differentiates back to the integrand. Why distractors fail: Option A misidentifies the derivative of $\cos x$. Option C is wrong because $\cos x$ and $-\cos x$ differ by a factor of $-1$, not a constant. Option D is simply incorrect.

Answer: $\tan x + C$

Standard trig antiderivative: Since $\frac{d}{dx}[\tan x] = \sec^2 x$, we have $\int \sec^2 x\,dx = \tan x + C$. Why distractors fail: Option A is the derivative of $\sec^2 x$. Option C has an incorrect sign. Option D would differentiate to $\sec x \tan x$, not $\sec^2 x$.

Answer: $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$

Recall the power rule for antiderivatives: The power rule states $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Why the correct answer works: Option B matches the formula exactly: increase the exponent by 1 and divide by the new exponent. Why distractors fail: Option A is the derivative power rule, not the antiderivative rule. Option C subtracts 1 from the exponent instead of adding. Option D divides by the original exponent without adjusting it.

Answer: $-\dfrac{1}{2x^2} + C$

Apply the power rule with a negative exponent: $\int x^{-3}\,dx = \dfrac{x^{-2}}{-2} + C = -\dfrac{1}{2x^2} + C$. Why the correct answer works: Option A correctly increases the exponent from $-3$ to $-2$ and divides by $-2$. Why distractors fail: Option B divides by the old exponent $-3$ instead of the new exponent $-2$. Option C has the wrong sign. Option D is the derivative of $x^{-3}$, not its antiderivative.

Answer: $\dfrac{x^5}{5} + \dfrac{3}{x} + C$

Rewrite and integrate: $\int x^4\,dx = \dfrac{x^5}{5}$. For the second term, $-3x^{-2}$: $\int -3x^{-2}\,dx = -3 \cdot \dfrac{x^{-1}}{-1} = \dfrac{3}{x}$. Why the correct answer works: Option A gives $\dfrac{x^5}{5} + \dfrac{3}{x} + C$, correctly handling the negative exponent and the double negative. Why distractors fail: Option B has the wrong sign on $3/x$ (a common error when the negative from the coefficient and the negative from the integration cancel). Option C is the derivative. Option D divides by $-3$ instead of $-1$.

Answer: $\dfrac{x^3}{3} + 2x - \ln x + C$

Simplify first by dividing each term by $x$: $\dfrac{x^3 + 2x - 1}{x} = x^2 + 2 - \dfrac{1}{x}$. Integrate term by term: $\int x^2\,dx = \dfrac{x^3}{3}$, $\int 2\,dx = 2x$, $\int -\dfrac{1}{x}\,dx = -\ln x$ (since $x > 0$, we can drop the absolute value). Why distractors fail: Option B integrates the original numerator without dividing by $x$. Option C integrates $-1/x$ as $-1/x$ (the original function, not the antiderivative). Option D is the simplified integrand, not its antiderivative.

Answer: $e^x + C$

Exponential antiderivative: The function $e^x$ is its own derivative and its own antiderivative: $\int e^x\,dx = e^x + C$. Why the correct answer works: Option B is correct because $\frac{d}{dx}[e^x] = e^x$. Why distractors fail: Option A incorrectly applies the polynomial power rule to $e^x$. Option C similarly misapplies the power rule. Option D simplifies to $x + C$, whose derivative is 1, not $e^x$.

Answer: $s(t) = 2t^3 - 2t^2 + t + 2$

Integrate velocity to get position: $s(t) = \int (6t^2 - 4t + 1)\,dt = 2t^3 - 2t^2 + t + C$. Use the initial condition: $s(0) = 2$ gives $0 - 0 + 0 + C = 2$, so $C = 2$. Why distractors fail: Option A omits the constant $C = 2$. Option C takes the derivative instead of the antiderivative. Option D uses $C = -2$ instead of $C = 2$.

Answer: $x^3 + 2x^2 - 5x + C$

Apply the power rule term by term: $\int 3x^2\,dx = x^3$, $\int 4x\,dx = 2x^2$, $\int (-5)\,dx = -5x$. Why the correct answer works: Combining gives $x^3 + 2x^2 - 5x + C$, which is Option A. Why distractors fail: Option B is the derivative, not the antiderivative. Option C fails to divide $4x^2$ by 2. Option D fails to divide $3x^3$ by 3.

Answer: A function $F(x)$ such that $F'(x) = f(x)$

Definition of antiderivative: An antiderivative of $f(x)$ is a function $F(x)$ whose derivative equals $f(x)$; that is, $F'(x) = f(x)$. Why the correct answer works: Option B states the defining property: differentiating $F$ gives back $f$. Why distractors fail: Option A reverses the relationship. Option C describes a definite integral (a number, not a function family). Option D would make $F$ a second antiderivative of $f$.

Answer: $\ln|x| + C$

The special case $n = -1$: The power rule for antiderivatives does not apply when $n = -1$ because it would produce division by zero. Instead, $\int x^{-1}\,dx = \ln|x| + C$. Why the correct answer works: Option B uses the absolute value, which is necessary because $\ln x$ is defined only for $x > 0$, while $1/x$ is defined for all $x \neq 0$. Why distractors fail: Option A recognizes the power rule fails but doesn't provide the correct alternative. Option C omits the absolute value, making it invalid for negative $x$. Option D is the derivative of $1/x$, not the antiderivative.

Answer: The derivative is $-\cos x$, so the student's answer is incorrect; the correct antiderivative is $\sin x + C$

Differentiate to check: $\frac{d}{dx}[-\sin x + C] = -\cos x \neq \cos x$. Identify the correct antiderivative: Since $\frac{d}{dx}[\sin x] = \cos x$, the correct answer is $\sin x + C$. Why distractors fail: Option A incorrectly states the derivative is $\cos x$. Option C incorrectly states the derivative is $\sin x$. Option D adds a spurious constant to the derivative.

Answer: $2\ln|x| + 3e^x + C$

Integrate each term: $\int \dfrac{2}{x}\,dx = 2\ln|x|$ and $\int 3e^x\,dx = 3e^x$. Why the correct answer works: Option B correctly uses $\ln|x|$ (with absolute value) and $e^x$. Why distractors fail: Option A omits the absolute value around $x$. Option C differentiates $2/x$ instead of integrating. Option D misapplies the power rule to $e^x$.

Answer: $x^4 - 3x^2 + 2x + 5$

Integrate $f'(x)$: $f(x) = \int (4x^3 - 6x + 2)\,dx = x^4 - 3x^2 + 2x + C$. Solve for $C$: $f(1) = 1 - 3 + 2 + C = C = 5$, so $C = 5$. Why distractors fail: Option B sets $C = 0$. Option C is the derivative of $f'(x)$, not the antiderivative. Option D uses $C = 3$, which would give $f(1) = 1 - 3 + 2 + 3 = 3 \neq 5$.

Answer: $F(x) = \dfrac{2^x}{\ln 2} + 6\ln x + \left(-\dfrac{2}{\ln 2}\right)$

Integrate each term: $\int 2^x\,dx = \dfrac{2^x}{\ln 2}$ and $\int \dfrac{6}{x}\,dx = 6\ln x$ (since $x > 0$). So $F(x) = \dfrac{2^x}{\ln 2} + 6\ln x + C$. Find $C$ from $F(1) = 0$: $F(1) = \dfrac{2}{\ln 2} + 6\ln 1 + C = \dfrac{2}{\ln 2} + 0 + C = 0$, so $C = -\dfrac{2}{\ln 2}$. Why distractors fail: Option B omits the constant (sets $C = 0$), giving $F(1) = 2/\ln 2 \neq 0$. Option C uses $2^x \ln 2$ (the derivative of $2^x$) instead of $2^x / \ln 2$. Option D incorrectly integrates $6/x$ as $6/x^2$.

Answer: $-\cos x + 3\sin x + C$

Recall trig antiderivatives: $\int \sin x\,dx = -\cos x + C$ and $\int \cos x\,dx = \sin x + C$. Combine: $\int (\sin x + 3\cos x)\,dx = -\cos x + 3\sin x + C$. Why distractors fail: Option B has the wrong sign on $\cos x$. Option C has the wrong sign on $3\sin x$. Option D has both signs reversed.

Answer: Because differentiating any constant gives zero, so infinitely many antiderivatives exist that differ by a constant

Derivatives of constants vanish: If $F'(x) = f(x)$, then $(F(x) + C)' = f(x)$ for any constant $C$, because the derivative of a constant is zero. Why the correct answer works: Option B correctly identifies that the family of antiderivatives differs only by an additive constant, hence we write $+C$ to represent all of them. Why distractors fail: Option A is false — antiderivatives need not pass through the origin. Option C invokes a nonexistent dimensional requirement. Option D confuses $C$ with limits of a definite integral.

Answer: $\int (x^5 - 7\cos x + 3e^x)\,dx$

Analyze each integrand: Option B is a sum of a power function, a trig function, and an exponential — each is directly integrable term by term using standard rules. Why distractors require more advanced techniques: Option A ($e^{x^2}$) has no elementary antiderivative. Option C ($x\sin(x^2)$) requires u-substitution with $u = x^2$. Option D requires the substitution $u = 1 + e^x$. Key takeaway: Sums and constant multiples of basic integrable functions can be handled term by term; compositions of functions generally cannot.

Answer: $T(t) = -3\cos t + 2t + 103$

Integrate $T'(t)$: $T(t) = \int (3\sin t + 2)\,dt = -3\cos t + 2t + C$. Apply the initial condition: $T(0) = -3\cos 0 + 0 + C = -3 + C = 100$, so $C = 103$. Why distractors fail: Option A sets $C = 100$, forgetting to account for $-3\cos(0) = -3$. Option C has the wrong sign on the cosine term. Option D uses $C = 97$, which would give $T(0) = -3 + 97 = 94 \neq 100$.

Answer: The family of all functions whose derivative is $f(x)$, differing by a constant

Meaning of the indefinite integral: The notation $\int f(x)\,dx$ denotes the general antiderivative: the collection of all functions $F(x) + C$ such that $F'(x) = f(x)$. Why the correct answer works: Option B captures both the idea of reversing differentiation and the family of solutions differing by a constant. Why distractors fail: Option A describes the derivative, not the antiderivative. Option C describes a (possibly improper) definite integral, not an indefinite one. Option D describes a limit at infinity.

Answer: It is correct

Check each term by differentiating: $\frac{d}{dx}\left[\dfrac{x^3}{3}\right] = x^2$ ✓. $\frac{d}{dx}[e^x] = e^x$ ✓. $\frac{d}{dx}[\cos x] = -\sin x$, and the original integrand has $-\sin x$ ✓. Why the correct answer works: All three terms check out. The student correctly handled the sign: $\int -\sin x\,dx = \cos x + C$ because $\frac{d}{dx}[\cos x] = -\sin x$. Why distractors fail: Option B incorrectly applies the power rule to $e^x$. Option C overlooks that the negative sign in $-\sin x$ and the negative in $\frac{d}{dx}[\cos x] = -\sin x$ cancel, making $\cos x$ correct. Option D forgets to divide by 3.

Answer: $\sin x + C$

Recall the trig antiderivative: Since $\frac{d}{dx}[\sin x] = \cos x$, the antiderivative of $\cos x$ is $\sin x + C$. Why the correct answer works: Option B is correct because differentiating $\sin x$ gives $\cos x$. Why distractors fail: Option A has an incorrect negative sign. Option C would differentiate to $-\sin x$. Option D would differentiate to $\sin x$, not $\cos x$.

Answer: The power rule formula $\int x^n\,dx = x^{n+1}/(n+1) + C$ does not apply when $n = -1$; the correct result is $\ln|x| + C$

Identify the restriction on the power rule: The power rule $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ explicitly requires $n \neq -1$. The correct antiderivative: When $n = -1$, we use the special case $\int x^{-1}\,dx = \ln|x| + C$. Why distractors fail: Option A tries to assign meaning to an invalid application of the formula. Option C incorrectly suggests $+C$ resolves division by zero. Option D is false — the integral exists as $\ln|x| + C$.