Integration Rules

Flashcard 1

Front: Power Rule

Back: \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for} \, n \neq -1

Flashcard 2

Front: Constant Rule

Back: \int a \, dx = ax + C

Flashcard 3

Front: Constant Multiple Rule

Back: \int a \cdot f(x) \, dx = a \int f(x) \, dx

Flashcard 4

Front: Sum Rule

Back: \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx

Flashcard 5

Front: Difference Rule

Back: \int (f(x) - g(x)) \, dx = \int f(x) \, dx - \int g(x) \, dx

Flashcard 6

Front: Exponential Function

Back: \int e^x \, dx = e^x + C

Flashcard 7

Front: Natural Logarithm

Back: \int \frac{1}{x} \, dx = \ln|x| + C

Flashcard 8

Front: Integral of \sin x

Back: \int \sin x \, dx = -\cos x + C

Flashcard 9

Front: Integral of \cos x

Back: \int \cos x \, dx = \sin x + C

Flashcard 10

Front: Integral of \sec^2 x

Back: \int \sec^2 x \, dx = \tan x + C

Flashcard 11

Front: Integral of \csc^2 x

Back: \int \csc^2 x \, dx = -\cot x + C

Flashcard 12

Front: Integral of \sec x \cdot \tan x

Back: \int \sec x \cdot \tan x \, dx = \sec x + C

Flashcard 13

Front: Integral of \csc x \cdot \cot x

Back: \int \csc x \cdot \cot x \, dx = -\csc x + C

Flashcard 14

Front: Integral of \frac{1}{\sqrt{1 - x^2}}

Back: \int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1} x + C

Flashcard 15

Front: Integral of \frac{1}{1 + x^2}

Back: \int \frac{1}{1 + x^2} \, dx = \tan^{-1} x + C

Flashcard 16

Front: Integral of \frac{1}{|x|\sqrt{x^2 - 1}}

Back: \int \frac{1}{|x|\sqrt{x^2 - 1}} \, dx = \sec^{-1} x + C

Flashcard 17

Front: Integral of \sinh x

Back: \int \sinh x \, dx = \cosh x + C

Flashcard 18

Front: Integral of \cosh x

Back: \int \cosh x \, dx = \sinh x + C

Flashcard 19

Front: Integral of \text{sech}^2 x

Back: \int \text{sech}^2 x \, dx = \tanh x + C

Flashcard 20

Front: Integration by Parts

Back: \int u \, dv = uv - \int v \, du

Flashcard 21

Front: Substitution Rule (u-substitution)

Back: \int f(g(x))g{\prime}(x) \, dx = \int f(u) \, du \quad \text{where} \, u = g(x)