Difficulty: Intermediate | Prerequisites: Differentiation rules (power, trig, chain rule), algebra of exponents and radicals.
Antidifferentiation is the reverse of differentiation: given a derivative, recover the original function. This is the gateway to integral calculus, and every later topic in the course (area under curves, volumes, differential equations) depends on it. You need fluency with exponent laws and trig derivatives before this will click. If you are shaky on rewriting radicals as fractional powers or on the derivatives of sec, csc and their cousins, revisit Chapter 3 first.
An antiderivative of f(x) is any function F(x) whose derivative is f(x). The most general antiderivative adds an arbitrary constant C. Finding antiderivatives means reversing the power rule and memorising the standard trig antiderivatives.
Antiderivative
A function F(x) such that F'(x) = f(x). In simple terms, it is the function you had before you differentiated.
General antiderivative
The family F(x) + C, where C is an arbitrary constant. Because the derivative of any constant is zero, every antiderivative of a given function differs from every other by at most a constant.
Indefinite integral (notation)
The symbol ∫ f(x) dx denotes the general antiderivative of f(x). The dx tells you the variable of integration.
Constant of integration (C)
The arbitrary constant added to every general antiderivative. Omitting C is a common mark-losing mistake on exams.
Power Rule for Antiderivatives
If n ≠ −1, then ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C. Think of it as: raise the exponent by one, then divide by the new exponent.
Rewriting before integrating
The technique of expressing radicals, reciprocals and products in exponent form so the power rule applies. For example, 1/x² = x⁻², √x = x^(1/2), and x√x = x^(3/2).
To antidifferentiate xⁿ, add 1 to the exponent and divide by the new exponent.
This reverses the differentiation step that brings the exponent down and reduces it by 1.
The rule does not apply when n = −1 (that case gives ln|x| + C, covered later).
Before applying the power rule, rewrite every term so it looks like a constant times xⁿ.
1/x² becomes x⁻²
2/√x becomes 2x^(−1/2)
3√(x⁵) becomes 3x^(5/2)
Rewrite as: x⁻² + 2x^(−1/2) − 3x^(5/2)
Antidifferentiate term by term:
∫ x⁻² dx = x⁻¹/(−1) = −1/x
∫ 2x^(−1/2) dx = 2 · x^(1/2)/(1/2) = 4√x
∫ 3x^(5/2) dx = 3 · x^(7/2)/(7/2) = (6/7)x^(7/2)
Result: F(x) = −1/x + 4√x − (6/7)x^(7/2) + C
∫ cos t dt = sin t + C
∫ sin t dt = −cos t + C
∫ sec² t dt = tan t + C
∫ sec t tan t dt = sec t + C
∫ csc² t dt = −cot t + C
∫ csc t cot t dt = −csc t + C
These follow directly from reversing the standard derivative table.
Antidifferentiate term by term:
∫ 3 sin t dt = −3 cos t
∫ 4 cos t dt = 4 sin t
∫ 5 sec² t dt = 5 tan t
∫ 6 sec t tan t dt = 6 sec t
Result: G(t) = −3 cos t + 4 sin t + 5 tan t + 6 sec t + C
The antiderivative of a sum is the sum of the antiderivatives.
A constant factor pulls through: ∫ k · f(x) dx = k · ∫ f(x) dx.
These properties let you handle each term independently.
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)\int \cos t \, dt = \sin t + C\int \sin t \, dt = -\cos t + C\int \sec^2 t \, dt = \tan t + C\int \sec t \tan t \, dt = \sec t + CAntiderivatives underpin every problem where you know a rate of change and need the accumulated quantity. Engineers use them to go from a known force profile to velocity and displacement. Economists integrate marginal cost to recover total cost functions. Any time a sensor gives you a rate (flow rate, speed, growth rate), the antiderivative tells you the total.
Students often forget to add C. Every general antiderivative requires it. If a problem asks for the "most general" antiderivative and you omit C, expect to lose marks.
Students sometimes apply the power rule to n = −1 and write x⁰/0, which is undefined. The case n = −1 is a special case that gives ln|x| + C.
Confusing the sign on trig antiderivatives is extremely common. The antiderivative of sin t is −cos t (negative), not +cos t. Differentiate your answer to check.
Students forget to rewrite radicals and fractions as powers of x before integrating. You cannot apply the power rule to 1/x² directly; you need x⁻² first.
⚠️ "Find the most general antiderivative" is bread-and-butter exam material. Expect at least one problem requiring you to rewrite terms in power form before integrating.
⚠️ Trig antiderivatives appear frequently. Memorise the table; there is no shortcut.
⚠️ Always verify by differentiating your answer. If F'(x) ≠ f(x), something went wrong.
⚠️ Problems that combine algebraic and trig terms in one function test whether you can handle both in sequence.
True or false: The antiderivative of x⁴ is x⁵. (False, it is x⁵/5 + C.)
Fill in the blank: ∫ cos t dt = ______ . (sin t + C)
True or false: ∫ x⁻¹ dx = x⁰/0 + C. (False, the power rule does not apply when n = −1.)
Fill in the blank: To antidifferentiate 2/√x, first rewrite it as ______ . (2x^(−1/2))
True or false: If F(x) and G(x) are both antiderivatives of f(x), then F(x) − G(x) is a constant. (True.)
Q: Find the most general antiderivative of f(x) = 5x⁴ − 3x² + 7.
A: F(x) = x⁵ − x³ + 7x + C
Q: Find the most general antiderivative of h(x) = 4/√x − 1/x².
A: Rewrite as 4x^(−1/2) − x⁻². Then H(x) = 4 · x^(1/2)/(1/2) − x⁻¹/(−1) + C = 8√x + 1/x + C.
Q: Find the most general antiderivative of g(t) = 2 cos t − 3 sec² t.
A: G(t) = 2 sin t − 3 tan t + C
Q: If F'(x) = 6x² and F(1) = 5, find F(x).
A: The general antiderivative is F(x) = 2x³ + C. Using F(1) = 5: 2(1)³ + C = 5, so C = 3. Therefore F(x) = 2x³ + 3.
Q: A student writes ∫ (sin t + sec t tan t) dt = cos t + sec t + C. What is wrong?
A: The antiderivative of sin t is −cos t, not +cos t. The correct answer is −cos t + sec t + C.
This material connects directly to initial value problems (§3.9 continued), where you use a given condition to solve for C. It also sets up definite integrals and the Fundamental Theorem of Calculus (Chapter 5), which formalises the link between antiderivatives and area. Trig antiderivatives reappear in integration techniques (substitution, parts) later in Calculus II.
Antiderivative, indefinite integral, reverse power rule, general antiderivative, constant of integration, trig antiderivatives, integration, ∫ f(x) dx, power rule for integrals, rewriting radicals as powers, MTH 132, Calculus I, Section 3.9