Unit 6 · Topic 6.8

6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

An indefinite integral ∫ f(x) dx stands for every antiderivative of f, written F(x) + C. You find the basic ones by running derivative rules backward, after rewriting expressions into forms the rules can handle.

Key terms

antiderivative
indefinite integral
constant of integration (+C)
power rule
ln|x|

Notation and the + C

∫ f(x) dx = F(x) + C means F′(x) = f(x), and C is any constant. The ∫ sign and dx go together: dx tells you the variable you're integrating with respect to. The + C is there because a constant disappears when you differentiate, so you can't know it from f alone.

An indefinite integral is a family of functions. A definite integral is a number. They're linked by the Fundamental Theorem (6.7).

Basic rules (each one is a derivative rule backward)

Integral	Result	Check by differentiating
∫ k dx	kx + C	(kx)′ = k
∫ xⁿ dx, n ≠ −1	xⁿ⁺¹/(n + 1) + C	power rule
∫ eˣ dx	eˣ + C	(eˣ)′ = eˣ
∫ aˣ dx (a > 0, a ≠ 1)	aˣ/ln a + C	(aˣ)′ = aˣ ln a
∫ sin x dx	−cos x + C	(−cos x)′ = sin x
∫ cos x dx	sin x + C	(sin x)′ = cos x
∫ sec² x dx	tan x + C	(tan x)′ = sec² x
∫ csc² x dx	−cot x + C	(−cot x)′ = −csc² x
∫ sec x tan x dx	sec x + C	(sec x)′ = sec x tan x
∫ csc x cot x dx	−csc x + C	(−csc x)′ = −csc x cot x
∫ 1/(1 + x²) dx	arctan x + C	(arctan x)′ = 1/(1 + x²)
∫ 1/√(1 − x²) dx	arcsin x + C	(arcsin x)′ = 1/√(1 − x²)

Rewrite before you integrate

The rules work term by term on sums, and constants come out front: ∫ [k·f(x) ± g(x)] dx = k∫ f(x) dx ± ∫ g(x) dx. Most of the work is algebra that turns the integrand into a sum of basic pieces:

Roots to exponents: √x = x^(1/2), 1/∛x = x^(−1/3).
Fractions over a single power: (x² + 3)/x = x + 3x⁻¹.
Expand products: (x + 2)² = x² + 4x + 4.
Use identities: tan² x = sec² x − 1.

Why the power rule skips n = −1, and a note on what can't be done

Using the power rule on x⁻¹ would divide by n + 1 = 0. That case has its own answer: ln|x| + C. The absolute value lets the result work for negative x too.

Some functions, such as e^(−x²), sin(x²) and (sin x)/x, have antiderivatives that can't be written with the usual algebraic, exponential, log and trig functions. The antiderivatives exist (6.4), but there's no formula to find. On the exam, those integrals come with a calculator or an accumulation function like ∫₀ˣ sin(t²) dt.

You can always check an antiderivative by differentiating it. If you get back the integrand, you're right.

Worked examples

Try each one yourself first, then open the solution.

Example 1
Term by term
Find ∫ (4x³ − 6√x + 5/x − 2) dx.
Show the solutionHide the solution
1. Step 1: Rewrite: 4x³ − 6x^(1/2) + 5x⁻¹ − 2.
2. Step 2: ∫ 4x³ dx = x⁴.
3. Step 3: ∫ 6x^(1/2) dx = 6 · x^(3/2)/(3/2) = 4x^(3/2).
4. Step 4: ∫ 5x⁻¹ dx = 5 ln|x|, and ∫ 2 dx = 2x.
5. Step 5: Combine and add one constant.
Answer: x⁴ − 4x^(3/2) + 5 ln|x| − 2x + C
Example 2
Rewrite first
Find ∫ (x² + 1)²/x² dx.
Show the solutionHide the solution
1. Step 1: Expand the top: (x² + 1)² = x⁴ + 2x² + 1.
2. Step 2: Divide each term by x²: x² + 2 + x⁻².
3. Step 3: Integrate: x³/3 + 2x + x⁻¹/(−1) = x³/3 + 2x − 1/x.
4. Step 4: Check: the derivative of x³/3 + 2x − 1/x is x² + 2 + 1/x², which matches.
Answer: x³/3 + 2x − 1/x + C
Example 3
Using an initial condition
f′(x) = 6x² − sin x and f(0) = 3. Find f(x).
Show the solutionHide the solution
1. Step 1: Antidifferentiate: f(x) = 2x³ + cos x + C (since the antiderivative of −sin x is cos x).
2. Step 2: Use f(0) = 3: 0 + cos 0 + C = 3, so 1 + C = 3 and C = 2.
3. Step 3: Trap: plugging in gives cos 0 = 1, not 0. Students who treat cos 0 as 0 get C = 3, which is wrong.
Answer: f(x) = 2x³ + cos x + 2

Common mistakes

Leaving off + C on an indefinite integral. On the exam that can cost a point.
Sign errors with trig: ∫ sin x dx = −cos x + C, not cos x + C.
Integrating a product or quotient piece by piece, as in ∫ x(x + 1) dx = (x²/2)(x²/2 + x). Expand first: ∫ (x² + x) dx = x³/3 + x²/2 + C.
Using the power rule on 1/x and writing x⁰/0.

On the exam

Antiderivative questions are common in the no-calculator multiple-choice section. Differentiate your answer, or a choice, to check it quickly.
In free response, if a function is defined by f′ and one value, find C right away and write the full function before using it.

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