Antidifferentiation and Basic Integration Rules

A collection of flashcards covering key concepts related to antidifferentiation and basic integration rules.

Antidifferentiation

The process of finding a function whose derivative is the given function.

Basic Integration Rule 1

$\frac{d}{dx}[C] = 0$ and $\int 0 \,dx = C$.

Basic Integration Rule 2

$\frac{d}{dx}[kx] = k$ and $\int k \,dx = kx + C$.

Basic Integration Rule 3

$\frac{d}{dx}[x^n] = nx^{n-1}$ and $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C,\text{ if } n \neq -1$; $\int x^{-1} \,dx = \ln x + C$.

Basic Integration Rule 4

$\frac{d}{dx}[kf(x)] = kf'(x)$ and $\int kf(x) \,dx = k \int f(x) \,dx.$

Basic Integration Rule 5

$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ and $\int [f(x) \pm g(x)] \,dx = \int f(x) \,dx \pm \int g(x) \,dx.$

Basic Integration Rule 6

$\frac{d}{dx}[e^x] = e^x$ and $\int e^x \,dx = e^x + C.$

Rewriting before Integrating

A technique used in integration that often simplifies the integral by rewriting the integrand.

Example of Antidifferentiation 1

$\int \sin(x^2) \,dx$.

Example of Antidifferentiation 2

$\int (\sin(\pi x) - \sin(3x)) \,dx$.

Example of Antidifferentiation 3

$\int \cos(4x) \,dx$.

Example of Antidifferentiation 4

$\int \tan(2x) \sec(2x) \,dx$.

Example of Antidifferentiation 5

$\int \sqrt{x^3}(x - 4) \,dx$.

Example of Antidifferentiation 6

$\int \left(\frac{\sin(x)}{1 - \sin^2(x)}\right) \,dx$.

Example of Antidifferentiation 7

$\int \left(x + 1 \sqrt{x}\right) \,dx$.

Example of Antidifferentiation 8

$\int (2\cos^2(x) - 1) \,dx$.