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1. Basic Integration Rules

Constant Rule: $\int k \, dx = kx + C$
Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$
Constant Multiple Rule: $\int k f(x) \, dx = k \int f(x) \, dx$
Sum/Difference Rule: $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$
Reciprocal Rule: $\int \frac{1}{x} \, dx = \ln|x| + C$

2. Exponential and Logarithmic Integrals

$\int e^x \, dx = e^x + C$
$\int a^x \, dx = \frac{a^x}{\ln a} + C$, for $a > 0, a \neq 1$
$\int \ln x \, dx = x \ln x - x + C$
$\int \log_a x \, dx = \frac{1}{\ln a} (x \ln x - x) + C$

3. Trigonometric Integrals

$\int \sin x \, dx = -\cos x + C$
$\int \cos x \, dx = \sin x + C$
$\int \sec^2 x \, dx = \tan x + C$
$\int \csc^2 x \, dx = -\cot x + C$
$\int \sec x \tan x \, dx = \sec x + C$
$\int \csc x \cot x \, dx = -\csc x + C$
$\int \tan x \, dx = \ln|\sec x| + C = -\ln|\cos x| + C$
$\int \cot x \, dx = \ln|\sin x| + C$
$\int \sec x \, dx = \ln|\sec x + \tan x| + C$
$\int \csc x \, dx = -\ln|\csc x + \cot x| + C$

4. Inverse Trigonometric Integrals

$\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C$
$\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$
$\int \frac{1}{x\sqrt{x^2 - a^2}} \, dx = \frac{1}{a} \operatorname{arcsec}\left(\frac{|x|}{a}\right) + C$

5. Integration Techniques

Integration by Substitution (u-Substitution)

If $u = g(x)$, then $du = g'(x) \, dx$.

$\int f(g(x))g'(x) \, dx = \int f(u) \, du$

Integration by Parts

$\int u \, dv = uv - \int v \, du$

Choose $u$ using LIATE (Logarithmic, Inverse Trig, Algebraic, Trigonometric, Exponential) for easier integration of $dv$.

Trigonometric Substitution

For $\sqrt{a^2 - x^2}$: Let $x = a \sin \theta$, $dx = a \cos \theta \, d\theta$
For $\sqrt{a^2 + x^2}$: Let $x = a \tan \theta$, $dx = a \sec^2 \theta \, d\theta$
For $\sqrt{x^2 - a^2}$: Let $x = a \sec \theta$, $dx = a \sec \theta \tan \theta \, d\theta$

Partial Fraction Decomposition

Used for rational functions $\frac{P(x)}{Q(x)}$ where degree of $P(x)$ is less than degree of $Q(x)$.

Distinct Linear Factors: $\frac{A}{ax+b} + \frac{B}{cx+d}$
Repeated Linear Factors: $\frac{A}{ax+b} + \frac{B}{(ax+b)^2}$
Irreducible Quadratic Factors: $\frac{Ax+B}{ax^2+bx+c}$

6. Definite Integrals

Fundamental Theorem of Calculus (Part 1)

If $F'(x) = f(x)$, then $\int_a^b f(x) \, dx = F(b) - F(a)$

Fundamental Theorem of Calculus (Part 2)

$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$

$\frac{d}{dx} \int_{g(x)}^{h(x)} f(t) \, dt = f(h(x))h'(x) - f(g(x))g'(x)$

Properties of Definite Integrals

$\int_a^a f(x) \, dx = 0$
$\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$
$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$
If $f(x) \ge 0$ on $[a,b]$, then $\int_a^b f(x) \, dx \ge 0$

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