6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Use the power-rule antiderivative. For any n ≠ −1, ∫ x^n dx = x^(n+1)/(n+1) + C. Here n = 3, so ∫ x^3 dx = x^4/4 + C, where C is the constant of integration (CED FUN-6.C.1). Check it by differentiating: (d/dx)(x^4/4 + C) = x^3, so it’s correct. This is a basic polynomial antiderivative example you’ll see on the AP (Topic 6.8—FUN-6.C and polynomial antiderivatives). For more examples and quick review, see the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and practice lots of problems at (https://library.fiveable.me/practice/ap-calculus).

The basic formula is: ∫ f(x) dx = F(x) + C, where F′(x) = f(x). That means an indefinite integral is any antiderivative F of f, plus a constant C. Why +C? Because differentiation kills constants: if F′(x)=f(x), then (F(x)+5)′ = F′(x) = f(x). So all antiderivatives differ only by a constant, and the +C records that whole family. Think practically: using the power-rule antiderivative, ∫ x^n dx = x^{n+1}/(n+1) + C (n ≠ −1). You must include +C on AP free-response when they ask for an indefinite integral (CED FUN-6.C.1). For definite integrals you evaluate F(b)−F(a) so the constants cancel. For more review and examples tied to the AP curriculum, see the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Use a derivative when you’re given a function and want its instantaneous rate of change (slope)—that’s f′(x). Use an antiderivative (indefinite integral) when you’re given a rate and want the original function whose derivative is that rate. In CED terms: if F′(x) = f(x), then ∫f(x) dx = F(x) + C (constant of integration). Quick rules to decide: - Problem asks for slope, tangent, acceleration, or rate of change at a point → take a derivative. - Problem asks for the original quantity from a rate, an accumulation, or “find all functions whose derivative is …” → find an antiderivative / evaluate an indefinite integral. - Use derivative rules backward (power rule, trig, e^x, ln|x|) or u-substitution for antiderivatives. Remember many functions have no elementary antiderivative (you’ll see special functions or numeric methods). Unit 6 on the AP exam focuses on this (integration & accumulation), so practice both types (study guide: https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3; practice problems: https://library.fiveable.me/practice/ap-calculus).

An antiderivative of f is any function F whose derivative is f (so F′(x)=f(x)). An indefinite integral, written ∫f(x) dx, denotes the entire family of antiderivatives: ∫f(x) dx = F(x) + C (the constant of integration). So “antiderivative” names a particular function; “indefinite integral” names the set of all those functions and the operation of finding them (FUN-6.C.1). Practically: when you solve F′(x)=f(x) you produce an antiderivative F; when you write ∫f(x) dx you should include “+ C” because there are infinitely many antiderivatives differing by a constant. Remember some functions have no elementary (closed-form) antiderivative (FUN-6.C.3), and you’ll use rules from differentiation in reverse (power rule, trig, ln|x|, e^x, linearity; and sometimes u-substitution). For more AP-aligned review see the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and extra practice problems (https://library.fiveable.me/practice/ap-calculus).

Think of an indefinite integral as “find a function whose derivative is this.” Using linearity and the power-rule antiderivative (CED FUN-6.C): Step 1—split by linearity: ∫(2x + 5) dx = ∫2x dx + ∫5 dx. Step 2—use constant multiple and power rule: ∫2x dx = 2·∫x dx = 2·(x^2/2) = x^2. (Power rule: ∫x^n dx = x^(n+1)/(n+1) for n ≠ −1.) Step 3—integrate the constant: ∫5 dx = 5x. Step 4—combine and add the constant of integration (CED FUN-6.C.1): ∫(2x + 5) dx = x^2 + 5x + C. Quick check: differentiate x^2+5x+C → 2x+5. That’s the antiderivative. For more on basic rules and notation, see the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and extra practice (https://library.fiveable.me/practice/ap-calculus).

Good question—the +C is there because antiderivatives aren’t unique. If F′(x) = f(x), then (F(x) + any constant)′ = F′(x) = f(x). So the indefinite integral ∫f(x) dx = F(x) + C (CED FUN-6.C.1). That C represents all possible vertical shifts (a “family” of antiderivatives). Why it matters: - Without extra info, you can only find the family, not one specific curve. - An initial condition (like F(a) = b) picks the right C; that’s common in physics and differential equations. - On the AP exam, when you evaluate an indefinite integral, include +C unless the problem gives an initial value or asks for a particular antiderivative. If you want more quick examples and practice, see the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Think of antiderivatives as “undoing” derivatives—so look at the form of the integrand and ask which derivative rule would have produced it. Quick checklist (use CED keywords): - Power rule reversed: for x^n (n ≠ −1) use ∫x^n dx = x^(n+1)/(n+1) + C. - Linearity: split sums/constants out (∫(a f + b g) = a∫f + b∫g). - Chain rule reversed → try u-substitution: if you see f(g(x)) times something like g'(x) (or a constant multiple), set u = g(x). Common cues: (ax+b) raised to a power, or 1/(1+ x^2) with x in numerator, etc. - Recognize standard pairs from the basic antiderivative table: e^x ↔ e^x, 1/x ↔ ln|x|, cos↔sin, sin↔−cos, sec^2↔tan, 1/√(1−x^2)↔arcsin, etc. - Product rule doesn’t reverse directly—use integration by parts when integrand looks like product of two functions (e.g., x·e^x). - If nothing fits, the antiderivative might be non-elementary (CED: many functions have no closed-form antiderivative). Always check by differentiating your result. Practice spotting patterns with the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and use lots of practice problems (https://library.fiveable.me/practice/ap-calculus)—AP exam expects familiarity with basic antiderivative forms and u-substitution.

It means there’s no antiderivative you can write using the usual elementary functions (polynomials, exponentials, logs, trig, inverse trig, etc.). In other words, no closed-form F(x) made from those pieces satisfies F′(x)=f(x). A classic example is f(x)=e^(−x^2); its antiderivative is the non-elementary error function erf(x). What that implies for you: you can’t expect to find a neat symbolic ∫f(x)dx using basic antiderivative rules or u-substitution. Instead you (a) express the result with a special function (like erf), (b) use a power series expansion, or (c) evaluate definite integrals numerically (Riemann sums, technology, or the FTC). On the AP exam you won’t be asked to produce non-elementary antiderivatives; focus on applying the basic antiderivative table, power rule, linearity, u-sub, and the FTC (CED FUN-6.C points). For a targeted review, see the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Think about derivatives reversed: an antiderivative F(x) satisfies F′(x) = f(x), and an indefinite integral is written ∫f(x) dx = F(x) + C (FUN-6.C.1). - ∫ sin x dx = −cos x + C because (−cos x)′ = sin x. - ∫ cos x dx = sin x + C because (sin x)′ = cos x. You can check each by differentiating your answer. Use linearity too: constants pull out and sums integrate term-by-term (FUN-6.C.2). Always include the constant of integration C for indefinite integrals. If you want more examples, a basic antiderivative table is in the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3). For extra practice problems across Unit 6, see (https://library.fiveable.me/ap-calculus/unit-6) and lots of practice questions at (https://library.fiveable.me/practice/ap-calculus).

Think of ∫f(x) dx as shorthand for “the family of antiderivatives of f.” Here’s what each part means: - ∫ is the integral sign—it tells you you’re finding an antiderivative (an indefinite integral), not a number (that’d be a definite integral with limits). - f(x) is the integrand—the function you want an antiderivative of. - dx indicates the variable of integration (x). It reminds you which variable you’re undoing differentiation with respect to. So ∫f(x) dx = F(x) + C, where F′(x) = f(x) and C is the constant of integration (FUN-6.C.1). Because derivatives rule antiderivatives, you use derivative rules backwards (power rule, linearity, trig, e^x, ln|x|, etc.; FUN-6.C.2). Remember: some functions don’t have elementary antiderivatives (you’ll see things like the error function)—that’s also on the CED (FUN-6.C.3). If you want a quick topic review, check the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3). For more practice, Fiveable has lots of problems (https://library.fiveable.me/practice/ap-calculus).

Quick check: differentiate your answer. If F(x) + C is your antiderivative for ∫f(x) dx, then F′(x) should equal f(x) on the domain. That’s the main rule in the CED (FUN-6.C.1–6.C.2). Practical checklist: - Differentiate your result and simplify—you should get the original integrand exactly. - Make sure you included +C (or note it); missing it is a common slip. - Watch domains: ln|x| versus ln x, and absolute values when needed. - For substitution answers, undo the u-sub to verify (differentiate F and apply chain rule). - For trig, exponential, and inverse-trig antiderivatives, compare to known table entries (power rule, e^x, sin/cos, sec^2 → tan, arcsin, etc.). - Remember some integrands have no elementary antiderivative (FUN-6.C.3)—don’t force a closed form. For extra practice and reference tables, check the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and more problems at (https://library.fiveable.me/practice/ap-calculus).

The antiderivative of e^x is e^x + C. That’s because an antiderivative F(x) of f(x) must satisfy F′(x) = f(x) (FUN-6.C.1). Since d/dx[e^x] = e^x for every x, e^x itself is a function whose derivative equals e^x—so any antiderivative is e^x plus an arbitrary constant C (the constant of integration). In notation: ∫ e^x dx = e^x + C. Why it “stays the same” is just that e^x is the unique (up to a constant) function that is unchanged by differentiation. Remember on the AP you should cite the antiderivative and include +C unless a definite integral is given. For extra practice on basic antiderivative rules see the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

The power-rule antiderivative says ∫ x^n dx = x^(n+1)/(n+1) + C, but that formula requires n ≠ −1 because you’d divide by zero when n = −1. For f(x) = 1/x we have n = −1, so the power rule fails. Instead use the fact that d/dx[ln|x|] = 1/x for all x ≠ 0. Therefore ∫ (1/x) dx = ln|x| + C, where the absolute value handles negative x (the derivative of ln|x| is 1/x on both sides). This is a basic antiderivative you should know for the AP exam (it’s in the FUN-6.C antiderivative rules). For extra practice and a quick review of rules and notation, check the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Short answer: yes—but with an important caveat. Power-rule for antiderivatives: for any real n ≠ −1, ∫ x^n dx = x^(n+1)/(n+1) + C. So you add 1 to the exponent and put n+1 in the denominator. Example: ∫ x^2 dx = x^3/3 + C. If there’s a constant factor, pull it out: ∫ 5x^4 dx = 5·x^5/5 = x^5 + C. Caveat: when n = −1, x^(−1) = 1/x, and the rule would divide by zero. Instead ∫ 1/x dx = ln|x| + C. Quick tips for remembering: think “reverse derivative”—differentiate x^(n+1)/(n+1) and you get x^n. Practice common antiderivatives on the AP (e^x, ln|x|, sin, cos, sec^2, inverse trig) and use linearity often. For a focused review, check the Topic 6.8 study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Think of integration as the reverse of differentiation: an antiderivative F(x) must satisfy F'(x) = f(x). For many simple f—polynomials, e^x, sin x, etc.—we have rules (power rule, linearity, trig and exponential rules, u-substitution) that give elementary antiderivatives. But some functions don’t match any combination of those elementary building blocks. Showing this rigorously requires higher theory (Liouville’s theorem), but the practical idea is: the class of elementary functions (polynomials, exponentials, logs, trig, algebraic combinations) isn’t closed under taking antiderivatives. So some integrals are “non-elementary”—they have antiderivatives that exist but are expressed with special functions (for example, the Gaussian integral leads to the error function erf(x); ∫e^{x^2} dx has no elementary antiderivative). On the AP exam you should: know the basic antiderivative table, use substitution, and recognize when you need a numerical or special-function answer rather than an elementary closed form. For a quick topic review, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3) and try practice problems (https://library.fiveable.me/practice/ap-calculus).