AP Calculus AB Quiz: Selecting Techniques For Antidifferentiation

What this quiz covers

This quiz focuses on Selecting Techniques For Antidifferentiation, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus AB.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Trigonometric substitution
2. Integration by parts
3. Substitution with $u=\ln x$ (correct answer)
4. Partial fraction decomposition
5. Power-reduction identities

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ 1/(x (ln x)²) dx, substitution with u = ln x is most appropriate because du = (1/x) dx directly matches the 1/x factor, simplifying the integral to ∫ 1/u² du. This substitution transforms the composite function into a basic power rule integral. The presence of ln x and its derivative 1/x in the integrand makes this a clear candidate for logarithmic substitution. While integration by parts might be tempting for products, it fails here as it would complicate the integral without simplifying the logarithmic term. Always look for inner functions whose derivatives appear in the integrand to identify substitution opportunities.

Question 2

1. Partial fraction decomposition
2. Integration by parts
3. Substitution with $u=1+\cos x$ (correct answer)
4. Trigonometric substitution
5. Polynomial long division

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand $\sin x / (1 + \cos x)$ has $\sin x$ in the numerator, which is the negative derivative of $\cos x$ in the denominator, suggesting substitution. Letting $u = 1 + \cos x$, $du = -\sin x \, dx$, gives $-\int \frac{du}{u} = -\ln|u| + C$. This simplifies the trig rational directly. Partial fractions might be tempting if misreading as rational in x, but the trig functions make it inapplicable. Identify substitutions in trig rationals where the numerator matches the derivative of part of the denominator.

Question 3

1. Integration by parts
2. Trigonometric substitution
3. Split into simpler terms and use basic rules (correct answer)
4. Partial fraction decomposition
5. Substitution with $u=1/x$

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ (1/x - 1/x²) dx, splitting into simpler terms and using basic rules is most appropriate because it becomes ∫ 1/x dx - ∫ x^{-2} dx, yielding ln|x| + 1/x + C. Each term is a standard integral. No advanced techniques are needed for this algebraic separation. While substitution with u = 1/x might be tempting, it fails as it complicates the integral without necessity. Always simplify expressions by splitting before applying integration rules.

Question 4

1. Substitution with $u=x^2-x$ (correct answer)
2. Partial fraction decomposition
3. Trigonometric identities then integrate
4. Integration by parts
5. Disk/washer volume method

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand features an exponential with a composite exponent and a polynomial factor that matches the derivative of that exponent, ideal for substitution. Setting u = x² - x gives du = (2x - 1) dx, transforming the integral into ∫ e^u du, which is simply e^u + C. This substitution captures the entire structure efficiently without needing decomposition or parts. Integration by parts might tempt due to the product form, but it would complicate things unnecessarily since the substitution directly simplifies the exponential composition. To spot substitution, check if a factor in the integrand is the derivative of the inner function of a composite.

Question 5

1. Integration by parts
2. Substitution with $u=1+e^{2x}$ (correct answer)
3. Partial fractions
4. Trigonometric substitution
5. Complete the square then substitute

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand $\frac{e^{2x}}{1 + e^{2x}}$ resembles a form where the numerator is related to the derivative of the denominator, suggesting substitution. Letting $u = 1 + e^{2x}$, $du = 2 e^{2x} \, dx$, gives $\frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| + C$. This substitution simplifies the exponential rational function directly. Integration by parts might tempt for the apparent product, but there's no clear product, making it inefficient compared to sub. Spot substitution in rational functions with exponentials when the numerator matches part of the denominator's derivative.

Question 6

1. Substitution with $u=x^3+7$ (correct answer)
2. Integration by parts
3. Trigonometric substitution
4. Partial fractions
5. Area of a semicircle

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is a rational function where the numerator x² is nearly the derivative of the denominator x³ + 7, which is 3x², making substitution ideal. Setting u = x³ + 7 gives du = 3x² dx, so the integral is (1/3) ∫ du/u = (1/3) ln|u| + C. This matches perfectly, simplifying the polynomial ratio. Partial fractions might tempt for rational functions, but it fails here as the denominator isn't factored into linears or quadratics suitable for decomposition. Check if the numerator is a scalar multiple of the denominator's derivative to prioritize substitution over other rational techniques.

Question 7

1. Integration by parts
2. Substitution with $u=x^2-16$ (correct answer)
3. Partial fraction decomposition
4. Power-reduction identities
5. Trigonometric substitution

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand x / √(x² - 16) has x in the numerator, half the derivative of x² - 16, making substitution appropriate. Setting u = x² - 16, du = 2x dx, gives (1/2) ∫ du / √u = √u + C. This handles the radical quadratic directly. Trigonometric substitution might be tempting for √(x² - a²), but sub is simpler here without introducing angles. Prioritize substitution for radicals when a linear factor matches the inner derivative.

Question 8

1. Partial fraction decomposition
2. Basic logarithm rule with linear substitution (correct answer)
3. Integration by parts
4. Trigonometric substitution
5. Disk method

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral $\int \frac{5}{2x - 3} \, dx$, the basic logarithm rule with linear substitution is most appropriate because it is a constant over a linear function, which integrates to $\frac{5}{2} \ln|2x - 3| + C$. A simple $u = 2x - 3$ substitution confirms $du = 2 \, dx$, adjusting the constant accordingly. This form directly matches the integral of $\frac{1}{u} \, du$ up to scalars. While partial fractions might be tempting for rational functions, it fails here as there's no need to decompose a single linear denominator. Always check if a rational integrand is already in its simplest form before considering decomposition.

Question 9

1. Partial fractions
2. Integration by parts
3. Recognize an inverse trigonometric derivative pattern (correct answer)
4. Trigonometric substitution
5. Polynomial long division

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral $\int \frac{2}{1 + x^2} \, dx$, recognizing an inverse trigonometric derivative pattern is most appropriate because it matches the derivative of $\arctan(x)$, yielding $2 \arctan(x) + C$. The form $\frac{1}{1 + x^2}$ is the standard pattern for arctan. Adjusting the constant 2 makes it straightforward. While trigonometric substitution might be tempting, it fails as it's unnecessary and more complex for this basic form. Always identify if the integrand resembles known derivatives of inverse trig functions for quick solutions.

Question 10

1. Integration by parts
2. Partial fraction decomposition
3. Algebraic manipulation using a trig identity (correct answer)
4. Trigonometric substitution
5. Long division then integrate

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is a product of sine and cosine with the same argument, which suggests using a trigonometric identity to simplify before integrating. The double-angle identity sin(2θ) = 2 sin θ cos θ allows rewriting sin(3x) cos(3x) as (1/2) sin(6x), leading to a straightforward integral of (1/2) ∫ sin(6x) dx. This algebraic manipulation fits perfectly for products of like trig functions. Integration by parts might be tempting for the product form, but it would lead to more complex terms without simplification, whereas the identity reduces it efficiently. Look for trig products that match known identities to simplify the integrand before choosing other methods.

Question 11

1. Algebraic manipulation (multiply by conjugate) then basic trig integrals (correct answer)
2. Partial fraction decomposition
3. Integration by parts
4. Substitution with $u=1+\sin x$
5. Trigonometric substitution

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ 1/(1 + sin x) dx, algebraic manipulation by multiplying by the conjugate then basic trig integrals is most appropriate because multiplying numerator and denominator by 1 - sin x yields (1 - sin x)/(cos² x) = sec² x - sec x tan x, which integrates to tan x - sec x + C. This rationalizes the trig expression. The form suggests conjugate multiplication for simplification. While substitution with u = 1 + sin x might be tempting, it fails as du = cos x dx doesn't directly help. Always consider multiplying by conjugates for trig rationals involving sine or cosine.

Question 12

1. Substitution with $u=\ln x$
2. Integration by parts (correct answer)
3. Trigonometric substitution
4. Partial fractions
5. Reverse the Chain Rule directly

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is ln x, a transcendental function without an elementary substitution or direct rule, making integration by parts the best choice. By setting u = ln x and dv = dx, it becomes ∫ u dv = uv - ∫ v du, yielding x ln x - ∫ x (1/x) dx = x ln x - x + C. This method effectively handles the logarithmic function multiplied implicitly by 1. Substitution with u = ln x might seem promising, but it fails because du = (1/x) dx, leaving no x to pair with, resulting in an incomplete transformation. Recognize integration by parts when the integrand is a product where one part is easy to integrate and the other to differentiate.

Question 13

1. Partial fraction decomposition
2. Substitution with $u=x^2+9$ (correct answer)
3. Integration by parts
4. Trigonometric substitution
5. Use a trig identity to reduce power

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand x √(x² + 9) has x in the numerator, which is half the derivative of x² + 9, suggesting substitution. Letting u = x² + 9, du = 2x dx, the integral becomes (1/2) ∫ √u du = (1/3) u^{3/2} + C. This substitution directly resolves the square root composition. Trigonometric substitution might be tempting for the √(x² + a²) form, but it's unnecessary here as the simple u-sub handles it without introducing trig functions. Recognize substitution when the integrand includes a radical of a quadratic and a factor matching its derivative.

Question 14

1. Complete the square then substitution (correct answer)
2. Partial fraction decomposition
3. Integration by parts
4. Trigonometric substitution
5. Disk method

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral $\int \frac{x + 4}{x^2 + 4x + 13}\, dx$, completing the square then substitution is most appropriate because the denominator becomes $(x + 2)^2 + 9$, allowing a split into integrals resembling $\ln$ and $\arctan$ forms. Rewrite the numerator as $\frac{1}{2} (2x + 4) + 2$ to match the derivative of the completed square. This transforms it into manageable parts. While trigonometric substitution might be tempting after completing the square, it fails as the direct method is simpler without trig identities. Always complete the square for quadratics that don't factor easily to reveal standard forms.

Question 15

1. Trigonometric substitution
2. Integration by parts
3. Basic logarithm rules after splitting the integral (correct answer)
4. Partial fraction decomposition
5. Use the Fundamental Theorem of Calculus Part 1

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is already split into a difference of fractions, each resembling the form for logarithmic integration after splitting. Integrating term by term gives 3 ln|x+2| - ln|x-1| + C, using basic log rules without further decomposition. This direct approach leverages the pre-split structure for simplicity. Partial fraction decomposition might seem tempting if one overlooks the existing form, but it's redundant here since the integral is already in integrable pieces. When an integrand is a sum or difference of simple rational terms, try integrating each directly before considering decomposition.

Question 16

1. Partial fraction decomposition
2. Integration by parts
3. Substitution with $u=2x+1$ (correct answer)
4. Trigonometric substitution
5. Geometric area interpretation

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ 4/√(2x + 1) dx from 1 to 4, substitution with u = 2x + 1 is most appropriate because du = 2 dx, so 2 ∫ u^{-1/2} du, yielding 4 √u + C. This simplifies the square root in the denominator. The linear inside the root with a matching differential makes it ideal. While trigonometric substitution might be tempting for roots, it fails as it's overkill for this simple form. Always use substitution for compositions where the inner function's derivative is present.

Question 17

1. Partial fraction decomposition
2. Integration by parts
3. Recognize an inverse trigonometric derivative pattern (correct answer)
4. Polynomial long division
5. Substitution with $u=4-x^2$

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ 1/√(4 - x²) dx, recognizing an inverse trigonometric derivative pattern is most appropriate because the form matches the derivative of arcsin(x/2), which is 1/√(4 - x²) after adjusting constants. This direct recognition allows for immediate antidifferentiation without additional steps. The integrand's square root of a quadratic in the denominator with a negative x² term signals this inverse sine pattern. While substitution with u = 4 - x² might seem tempting, it fails because the du = -2x dx introduces an x in the numerator that isn't present, requiring further adjustments. Always scan for standard integral forms like inverse trig functions when dealing with square roots of quadratics.

Question 18

1. Partial fraction decomposition
2. Algebraic simplification then basic power rule (correct answer)
3. Integration by parts
4. Trigonometric substitution
5. Substitution with $u=(x+1)^4$

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ (x+1)^4 / (x+1) dx, algebraic simplification then basic power rule is most appropriate because it reduces to ∫ (x+1)^3 dx, which expands or substitutes easily to (1/4)(x+1)^4 + C. Simplifying the fraction first reveals the polynomial. This avoids overcomplicating with other methods. While substitution with u = (x+1)^4 might be tempting, it fails as it doesn't address the denominator properly. Always cancel common factors in rational expressions before integrating.

Question 19

1. Integration by parts
2. Substitution with $u=\sin x$ (correct answer)
3. Partial fraction decomposition
4. Trigonometric substitution
5. Polynomial long division

Explanation: Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral $\int \frac{\cos x}{\sin x} \, dx$, substitution with $u = \sin x$ is most appropriate because $du = \cos x \, dx$, simplifying to $\int \frac{du}{u} = \ln|\sin x| + C$. This directly matches the derivative in the numerator. The form is a classic logarithmic integral via substitution. While integration by parts might be tempting, it fails as it unnecessarily complicates a simple substitution. Always substitute when the numerator is the derivative of the denominator in trig functions.

Question 20

1. Integration by parts
2. Trigonometric substitution
3. Substitution with $u=x^2+4$ (correct answer)
4. Partial fraction decomposition
5. Use a geometric area formula

Explanation: This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is a rational function where the numerator is a constant multiple of the derivative of the denominator, making substitution a natural fit. By setting u = x² + 4, du = 2x dx, so the integral becomes 3 ∫ du/u, which simplifies to 3 ln|u| + C. This direct match between the numerator and the derivative of the inner function allows for straightforward integration. A tempting distractor like trigonometric substitution might seem appealing due to the quadratic in the denominator, but it is unnecessary here as the form doesn't require converting to a trig identity for simplification. Always look for cases where the numerator resembles the derivative of the denominator to recognize substitution opportunities.