Basic Integration Formulas Review

Vocabulary flashcards covering fundamental indefinite integration formulas, trigonometric integrals, hyperbolic integrals, inverse forms, and substitution concepts.

Substitution Method

A technique used to rewrite an integral into a standard form using $u$-substitution to evaluate more complicated functions.

Constant Rule for Integration

$\int k\,dx = kx + C$ for any real number $k$.

Power Rule for Integration

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

Reciprocal Integration Formula

$\int \frac{1}{x}\,dx = \ln|x| + C$.

Exponential Integration Formula (Base $e$)

$\int e^x\,dx = e^x + C$.

Exponential Integration Formula (General Base)

$\int a^x\,dx = \frac{a^x}{\ln(a)} + C$ where $a > 0$ and $a \neq 1$.

Sine Integration Formula

$\int \sin(x)\,dx = -\cos(x) + C$.

Cosine Integration Formula

$\int \cos(x)\,dx = \sin(x) + C$.

Secant Squared Integration Formula

$\int \sec^2(x)\,dx = \tan(x) + C$.

Cosecant Squared Integration Formula

$\int \csc^2(x)\,dx = -\cot(x) + C$.

Secant-Tangent Integration Formula

$\int \sec(x)\tan(x)\,dx = \sec(x) + C$.

Cosecant-Cotangent Integration Formula

$\int \csc(x)\cot(x)\,dx = -\csc(x) + C$.

Tangent Integration Formula

$\int \tan(x)\,dx = \ln|\sec(x)| + C$.

Cotangent Integration Formula

$\int \cot(x)\,dx = \ln|\sin(x)| + C$.

Secant Integration Formula

$\int \sec(x)\,dx = \ln|\sec(x) + \tan(x)| + C$.

Cosecant Integration Formula

$\int \csc(x)\,dx = -\ln|\csc(x) + \cot(x)| + C$.

Hyperbolic Sine Integration Formula

$\int \sinh(x)\,dx = \cosh(x) + C$.

Hyperbolic Cosine Integration Formula

$\int \cosh(x)\,dx = \sinh(x) + C$.

Inverse Sine Integral Form

$\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \sin^{-1}\left(\frac{x}{a}\right) + C$ for $a > 0$.

Inverse Tangent Integral Form

$\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$ for $a > 0$.

Inverse Secant Integral Form

$\int \frac{1}{x\sqrt{x^2 - a^2}}\,dx = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C$ for $a > 0$.

Inverse Hyperbolic Sine Integral Form

$\int \frac{1}{\sqrt{a^2 + x^2}}\,dx = \sinh^{-1}\left(\frac{x}{a}\right) + C$ for $a > 0$.

Inverse Hyperbolic Cosine Integral Form

$\int \frac{1}{\sqrt{x^2 - a^2}}\,dx = \cosh^{-1}\left(\frac{x}{a}\right) + C$ for $x > a > 0$.