📝 Properties of indefinite integrals (24 MCQs)

📖 Explanation: The correct reasoning is that $2C$ is still an arbitrary constant since $C$ is arbitrary; if $C$ can be any real number, then $2C$ also can be any real number. Thus, $3\sin x + 2C$ is just a different way of writing 3\sin x + C', where C' = 2C. Option A is incorrect because the constant doesn't "absorb" the factor 2; it's the arbitrariness of $C$ that makes it valid. Option C is false; the factor matters in notation but not in the result set. Option D oversimplifies the property.

📖 Explanation: The family of all antiderivatives of $2x$ is $x^2 + C$, where $C$ is any real constant. The expression $x^2 + 3$ is just a specific member of this family with $C = 3$. Thus, both are valid antiderivatives. Option B is wrong because any real number can be a constant, not just $C$. Option C is false because $C$ can be any number, not just zero. Option D is incorrect because a specific constant is allowed; the arbitrary constant simply allows for all possibilities.

📖 Explanation: The integral of a linear combination is the linear combination of the integrals, by the properties of the indefinite integral. Since $\int f(x)\,dx = F(x) + C_1$ and $\int g(x)\,dx = G(x) + C_2$, then $\int [3f(x) - 2g(x)]\,dx = 3\int f(x)\,dx - 2\int g(x)\,dx = 3F(x) - 2G(x) + C$. The constant $C$ absorbs $3C_1 - 2C_2$. Option B misses the constant of integration. Option C has the wrong sign on 2. Option D incorrectly combines the functions.

📖 Explanation: The integral of a product is NOT equal to the product of integrals. There is no general product rule for indefinite integrals. Options A, B, and D are correct linearity properties: the integral of a sum/difference is the sum/difference of the integrals, and constants can be factored out. Option C represents a common misconception where students incorrectly apply linearity to multiplication, which is invalid. Integration does not distribute over multiplication; special techniques like integration by parts are needed for products.

📖 Explanation: Applying linearity and the power rule: $\int 3x^2\,dx = x^3 + C_1$. For $\int -\frac{2}{x^2}\,dx$, rewrite as $-2\int x^{-2}\,dx = -2\left(\frac{x^{-1}}{-1}\right) = 2x^{-1} = \frac{2}{x} + C_2$. And $\int 4\,dx = 4x + C_3$. Combining gives $x^3 + \frac{2}{x} + 4x + C$. Option B has the wrong sign on $2/x$. Option C omits the $4x$ term. Option D incorrectly keeps the coefficient 3 on $x^3$.

📖 Explanation: Rewrite $\frac{1}{x^2}$ as $x^{-2}$. Applying the power rule $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$ gives $\frac{x^{-1}}{-1} = -\frac{1}{x} + C$. The erroneous answer $\frac{1}{3x^3}$ would come from treating the denominator as if it were $x^3$ and then integrating incorrectly. Option B is wrong because the integral of $1/x^2$ is not logarithmic; the rule $\int 1/x\,dx = \ln|x| + C$ only applies when the exponent is exactly -1. Option D has the wrong sign.

📖 Explanation: Since $\sin^2 x + \cos^2 x = 1$, we have $\sin^2 x = 1 - \cos^2 x$. Thus, $F(x) = 1 - \cos^2 x + C = -\cos^2 x + (C+1)$. Let C' = C+1. Then F(x) = -\cos^2 x + C' = G(x). Both are valid antiderivatives of $2\sin x \cos x = \sin 2x$. Option A is wrong; they are related by a constant difference. Option B is false because a constant shift can make them equal for any $C$. Option D is incorrect because they both differentiate to $\sin 2x$.

📖 Explanation: In the expression $(x^2 + C_1) + (2x + C_2) = x^2 + 2x + (C_1 + C_2)$, the constants from different integrals are independent and should be denoted differently (e.g., $C_1$ and $C_2$). The final constant is just $C_1+C_2$, which is still an arbitrary constant, not $2C$ (unless one incorrectly assumes $C_1=C_2$). Option A is wrong because there is a flaw. Option C is irrelevant; integration is linear, not multiplicative. Option D is wrong; the constant can be any number, not always zero.

📖 Explanation: Using the trigonometric identity $\tan^2 x + 1 = \sec^2 x$. So $\int (\tan^2 x + 1)\,dx = \int \sec^2 x\,dx = \tan x + C$. Option B is wrong because $\sec^2 x$ is the integrand, not the integral; differentiating $\sec^2 x + C$ would not give $\sec^2 x$. Option C is the antiderivative of $\csc^2 x$, not $\sec^2 x$. Option D is the derivative of $\sec x$, not its antiderivative. Recognizing the identity is key to simplifying the integrand before integrating.

📖 Explanation: The derivative of any constant $C$ is 0, so the family of all functions whose derivative is zero is the set of all constant functions, i.e., $f(x) = C$. Option A is just one specific antiderivative (with C=0). Option C is the antiderivative of 1, not 0. Option D is another specific constant. This question tests Easy of the fact that the integral of zero is a constant, emphasizing the arbitrary nature of the constant of integration.

📖 Explanation: Rewrite the integrand as $\csc x \cot x$ because $\frac{\cos x}{\sin^2 x} = \frac{1}{\sin x} \cdot \frac{\cos x}{\sin x} = \csc x \cot x$. The derivative of $-\csc x$ is $\csc x \cot x$. Thus, the integral is $-\csc x + C$. Option B has the wrong sign. Option C is the antiderivative of $\sec x \tan x$ with a negative sign, which is not the given integrand. Option D is the antiderivative of $\sec x \tan x$ with the positive sign. It's important to recognize the derivative relationships of trigonometric functions.

📖 Explanation: The integral of a difference is the difference of the integrals: $\int [f(x) - g(x)]\,dx = \int f(x)\,dx - \int g(x)\,dx = [F(x) + C_1] - [G(x) + C_2] = F(x) - G(x) + (C_1 - C_2)$. Since $C_1$ and $C_2$ are arbitrary constants, their difference is also an arbitrary constant, which can be represented by $C$. Option B would represent the integral of a sum, not a difference. Option C misses the constant of integration. Option D is not a valid property.

📖 Explanation: Using the chain rule, $\frac{d}{dx}[\frac{1}{2}\ln|2x|] = \frac{1}{2} \cdot \frac{1}{2x} \cdot 2 = \frac{1}{2x}$. So the answer is correct. However, note that $\frac{1}{2}\ln|2x| + C = \frac{1}{2}(\ln|x| + \ln 2) + C = \frac{1}{2}\ln|x| + (\frac{1}{2}\ln 2 + C)$. Since $\frac{1}{2}\ln 2$ is a constant, it can be absorbed into $C$. Thus, the standard answer $\frac{1}{2}\ln|x| + C$ is also correct, and they differ by a constant. Option B is also correct but not the only correct; the question asks if the student's answer is correct. Option C is false. Option D is another correct form, but option A correctly identifies the derivative.

📖 Explanation: The constant multiple rule states $\int 5f(x)\,dx = 5\int f(x)\,dx = 5(\ln|x| + C_1) = 5\ln|x| + 5C_1$. Since $5C_1$ is just an arbitrary constant, we can write it as $C$. Option B is incorrect; $\ln|5x| = \ln 5 + \ln|x|$, which is not a constant multiple of $\ln|x|$. Option C is the intermediate step but not the most simplified or standard form. Option D is equivalent to $\ln|x|^5 = 5\ln|x|$ (by log properties), which is also a correct antiderivative, but option A is the simplest and most direct Easy of the rule. The question expects the direct Easy of the constant multiple rule.

📖 Explanation: We integrate term by term: $\int \sin x\,dx = -\cos x + C_1$ and $\int \cos x\,dx = \sin x + C_2$. Combining gives $-\cos x + \sin x + (C_1 + C_2) = \sin x - \cos x + C$. Option B is the derivative of $-\sin x + \cos x$, which is $-\cos x - \sin x$, not the original integrand. Option C is the integrand itself, not its antiderivative. Option D is the antiderivative of $-\sin x - \cos x$, not $\sin x + \cos x$. Recognizing the basic derivatives of sine and cosine is crucial.

📖 Explanation: The variable of integration in an indefinite integral is a dummy variable. $\int x^2\,dx = \frac{x^3}{3} + C$ and $\int t^2\,dt = \frac{t^3}{3} + C$. They represent the same rule: the antiderivative of the square of the independent variable is the cube of that variable divided by 3. The name of the variable does not affect the form of the result. Option A is wrong; the variable is just a placeholder. Option C is wrong; the arbitrary constant is always arbitrary regardless of the variable. Option D is false; they are comparable and identical in form.

📖 Explanation: Using substitution or the chain rule in reverse, if $u = 3x$, then $du = 3dx$, so $dx = du/3$. Thus, $\int \sec^2(3x)\,dx = \frac{1}{3}\int \sec^2 u\,du = \frac{1}{3}\tan u + C = \frac{1}{3}\tan(3x) + C$. Option B misses the necessary factor of $1/3$ from the derivative of $3x$. Option C is the answer one would get if they incorrectly ignored the chain rule. Option D is the derivative of $\sec(3x)$, not its antiderivative. This question tests the Easy of the chain rule in integration.

📖 Explanation: The integral of $e^x$ is $e^x + C_1$. The integral of $e^{-x}$ is $-e^{-x} + C_2$, because $\frac{d}{dx}[-e^{-x}] = e^{-x}$. Combining gives $e^x - e^{-x} + C$. Option B is the integrand itself. Option C incorrectly adds the constants. Option D misses the constant of integration. This question tests the integration of exponential functions, especially the negative exponent case, and reinforces the importance of the arbitrary constant.

📖 Explanation: This requires recognizing standard integral forms. $\int \frac{1}{1+x^2}\,dx = \tan^{-1} x + C_1$ and $\int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1} x + C_2$. Therefore, the integral is $\tan^{-1} x + \sin^{-1} x + C$. Option B has the wrong sign on the sine inverse term. Option C is just the original function. Option D is incorrect; it looks like the derivative of a quotient, not a standard integral. This question tests the ability to identify standard integrals from inverse trigonometric functions, a key concept in the chapter.

📖 Explanation: The function $f(x)$ is the derivative of the antiderivative. Differentiating $x^2 + \cos x + C$ gives $2x - \sin x + 0$. So, $f(x) = 2x - \sin x$. Option B is the derivative of $x^2 - \cos x$, which is not the given antiderivative. Option C is the antiderivative of $x^2 + \cos x$ with a different constant. Option D is the antiderivative of $x^2 - \sin x$. This tests the fundamental inverse relationship between differentiation and integration.

📖 Explanation: The student is using algebraic manipulation to rewrite the integrand in a more integrable form. This is allowed because the integral of a function is defined on the function itself; rewriting it using algebraic identities (like expanding a square) produces an equivalent expression. The sum/difference rule is then used to integrate the resulting polynomial. While options A, B, and C are used in the integration step, the initial step of simplifying the integrand is a valid algebraic technique, not a property of the integral itself. Option D best describes the general principle of simplifying the integrand before applying integration rules.

📖 Explanation: Since F'(x) = f(x), the derivative of $F$ is $f$. A maximum of $f(x)$ means f'(x) changes from positive to negative at $x=2$. But f'(x) = F''(x). Thus, the second derivative of $F$ changes from positive to negative at $x=2$, which is the definition of an inflection point. A local extremum of $F$ occurs where $f(x) = 0$, not where $f(x)$ has a maximum. Option B and C are wrong because a maximum of $f$ doesn't imply $f=0$. Option D is incorrect; we can determine it's an inflection point. This is an advanced HOTS question connecting graph behavior of a function and its antiderivative.

📖 Explanation: The most general antiderivative of $f(x)$ is $F(x) + C$, where $C$ is any constant. Adding any constant to $F(x)$ yields another antiderivative. However, adding $x$ (a non-constant function) does not generally yield an antiderivative of $f(x)$, because the derivative of $F(x) + x$ is $f(x) + 1$, which is only equal to $f(x)$ if $1 = 0$, which is false. Options A, B, and D add constants and are valid. This question tests the fundamental understanding that adding any constant to an antiderivative preserves the derivative, but adding a non-constant does not.

📖 Explanation: The correct antiderivative is $\sin^{-1} x + C$. The student incorrectly applied the power rule as if the integrand were $u^{-1/2}$, forgetting the chain rule factor from differentiating $u$. Differentiating the student's answer gives $\frac{d}{dx}[\frac{1}{2}(1-x^2)^{-1/2}] = \frac{1}{2} \cdot (-\frac{1}{2})(1-x^2)^{-3/2} \cdot (-2x) = \frac{x}{2(1-x^2)^{3/2}}$, which is not the original integrand. Option B is false; the power rule applies for $n \neq -1$. Option C is the correct rule but doesn't explain why the student's answer is wrong. Option D is not the main error; the algebra is flawed.