Left sum, right sum, midpoint sum, trapezoidal sum.

Approximation of area under curve using rectangles: $\int_a^b f(x)dx \approx \sum_{i=1}^n f(x_i^*) \Delta x$

where $\Delta x = \frac{b-a}{n}$

• Left: $\sum_{i=1}^n f(x_{i-1}) \Delta x$
• Right: $\sum_{i=1}^n f(x_i) \Delta x$
• Midpoint: $\sum_{i=1}^n f(\bar{x}_i) \Delta x$ where $\bar{x}_i = \frac{x_{i-1} + x_i}{2}$

Definite Integral as a Limit: ∫abf(x)dx=lim⁡...\int_a^b f(x)dx = \lim...∫ab​f(x)dx=lim...

$\int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$

FTC Part 1: d/dx [∫axf(t)dt\int_a^x f(t)dt∫ax​f(t)dt] = f(x)

$\frac{d}{dx}\left[\int_a^x f(t)dt\right] = f(x)$

Accumulation function: $g(x) = \int_a^x f(t)dt$, then g'(x) = f(x)

$A(x) = \int_a^x f(t)dt$ represents accumulated area/signed area from a to x.

$\int_a^b F'(x)dx = F(b) - F(a)$: net change in F over [a, b].

Properties: ∫ab+∫bc=∫ac\int_a^b + \int_b^c = \int_a^c∫ab​+∫bc​=∫ac​

$\int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx$

Properties: ∫ab=−∫ba\int_a^b = -\int_b^a∫ab​=−∫ba​

$\int_a^b f(x)dx = -\int_b^a f(x)dx$

FTC Part 2: ∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)∫ab​f(x)dx=F(b)−F(a)

$\int_a^b f(x)dx = F(b) - F(a)$

where F is any antiderivative of f.

$\int f(x)dx = F(x) + C$

where F'(x) = f(x) and C is constant of integration.

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

where u = g(x), du = g'(x)dx

When using u-substitution in definite integrals: $\int_a^b f(g(x))g'(x)dx = \int_{u(a)}^{u(b)} f(u)du$

For improper rational functions (numerator degree ≥ denominator):

1. Polynomial long division
2. Integrate resulting terms

$\int_a^\infty f(x)dx = \lim_{t \to \infty} \int_a^t f(x)dx$

$\int_a^b f(x)dx = \lim_{t \to b^-} \int_a^t f(x)dx$ (if f undefined at b)

$\int_a^\infty f(x)dx = \lim_{t \to \infty} \int_a^t f(x)dx$

$\int_a^b f(x)dx = \lim_{t \to b^-} \int_a^t f(x)dx$ (if f undefined at b)

• Converges: Limit exists and is finite
• Diverges: Limit doesn't exist or is infinite