Mathfolis

Antiderivatives

Unit 6 · Integration & Accumulation

Antiderivatives and Indefinite Integrals

The indefinite integral ∫ f(x) dx = F(x) + C is the most general antiderivative of f. Every differentiation rule has a corresponding antiderivative rule. The constant C is determined by an initial condition.

Power, Exponential, and Log

∫xn dx=xn+1n+1+C(n≠−1)∫1x dx=ln⁡∣x∣+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n\neq -1) \qquad \int \frac{1}{x}\,dx = \ln|x| + C
∫ex dx=ex+C\int e^x\,dx = e^x + C

Trigonometric

∫sin⁡x dx=−cos⁡x+C∫cos⁡x dx=sin⁡x+C\int \sin x\,dx = -\cos x + C \qquad \int \cos x\,dx = \sin x + C
∫sec⁡2x dx=tan⁡x+C∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec^2 x\,dx = \tan x + C \qquad \int \sec x\tan x\,dx = \sec x + C

Inverse Trig

∫11−x2 dx=arcsin⁡x+C∫11+x2 dx=arctan⁡x+C\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C \qquad \int \frac{1}{1+x^2}\,dx = \arctan x + C
AP Tip: Rewrite radicals and negative exponents as powers before integrating: √x = x^{1/2}, 1/x² = x^{-2}.
Type 1

Basic Antiderivative Rules

Apply the power, exponential, and trig rules term by term to find indefinite integrals.

Example 1

Find the indefinite integral.

∫(4x3−6x+2x+5ex)dx\int \left(4x^3 - 6x + \frac{2}{x} + 5e^x\right)dx
Example 2

Find the indefinite integral.

∫(cos⁡x+31+x2)dx\int \left(\cos x + \frac{3}{1+x^2}\right)dx
Example 3

Find the indefinite integral. (Rewrite first.)

∫(x+1x)dx\int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)dx

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Type 2

Initial Value Problems

Find the general antiderivative, then use the initial condition to solve for C.

Example 4

f′(x) = 3x² − 4x + 1 and f(0) = 2. Find f(x).

Example 5

f′(x) = 6x² − 4x + 1 and f(1) = 3. Find f(x).

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