Fundamental Calculus

Antiderivative¶

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

Example

\(\int x^3\ dx = \frac{x^4}{4} + C\)

Definition of Definite Integral¶

\(\int^b_a f(x) dx = \lim\limits_{n \to \infty} \sum\limits^n_{k=1} f(x^*_k) \cdot \Delta x\)

• Left point：\(x_k^* = a + (k-1)\Delta x\)
• Right point：\(x_k^* = a + k\Delta x\)
• Middle point：\(x_k^* = a + \left(k - \frac{1}{2}\right)\Delta x\)

Definite Integral¶

\(\int^b_a f(x) dx = \lim\limits_{n \to \infty} \sum\limits^n_{k=1} f(a + k \cdot \dfrac{b-a}{n}) \cdot \dfrac{b-a}{n}\)

Left Riemann Sum¶

\(\int^b_a f(x) dx = \lim\limits_{n \to \infty} \sum\limits^n_{k=1} f(a + (k-1) \cdot \dfrac{b-a}{n}) \cdot \dfrac{b-a}{n}\)

Right Riemann Sum¶

\(\int^b_a f(x) dx = \lim\limits_{n \to \infty} \sum\limits^n_{k=1} f(a + k \cdot \dfrac{b-a}{n}) \cdot \dfrac{b-a}{n}\)

Midpoint Riemann Sum¶

\(\int^b_a f(x) dx = \lim\limits_{n \to \infty} \sum\limits^n_{k=1} f(a + (k-\frac12) \cdot \dfrac{b-a}{n}) \cdot \dfrac{b-a}{n}\)

Trapezoidal Sum¶

\(\int^b_a f(x)dx = \lim\limits_{n \to \infty} \dfrac{b-a}{2n}[f(a) + 2\sum\limits^{n-1}{k=1}f(a + k \cdot \dfrac{b-a}{n} + f(b))]\)

\(\iff\)

\(\int^b_a f(x) dx = \lim\limits_{n \to \infty} \sum\limits^n_{k=1} \dfrac{f(a + (k-1) \cdot \frac{b-a}n + f(a+k \cdot \frac{b-a}n))}{2} \cdot \dfrac{b-a}n\)

Basic Integration Formulas¶

• \(\boxed{\int dx = x + C}\)
• \(\boxed{\int k\ dx = kx + C}\)
• \(\boxed{\int kf(x)dx = k \int f(x)dx}\)
• \(\boxed{\int[f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx}\)
• \(\boxed{\int x^n dx = \frac1{n+1} x^{n+1} + C}\) & \(\boxed{\int \frac1x dx = \ln |x| + C}\)
• \(\boxed{\int a^x dx = (\frac1{\ln a}) a^x + C}\)
• \(\int e^x dx = e^x + C\)
• \(\boxed{\int \sin x dx = - \cos x + C}\)
• \(\boxed{\int \cos x dx = \sin x + C}\)
• \(\boxed{\int \sec^2 x\ dx = -\tan x + C}\)
• \(\int \csc^2 x\ dx = -\cot x + C\)
• \(\int \sec x \tan x \ dx = \sec x + C\)
• \(\int \csc x \cot x \ dx = - \csc x + C\)
• \(\int \dfrac{dx}{\sqrt{1-x^2}} = \arcsin x + C\)
• \(\boxed{\int \dfrac{dx}{1+x^2}} = \arctan x + C\)
• \(\int \dfrac{dx}{\sqrt{a^2-x^2}} = \arcsin \dfrac ac + C\)
• \(\int \dfrac{dx}{a^2+x^2} = \dfrac 1a \arctan \dfrac xa + C\)