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Fundamental calculus

Antiderivative

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

Note

ff: integrand

dxdx: variable of integration

F(x)F(x): an antiderivative of f(x)f(x)

CC: constant of integration

Example

∫x3 dx=x44+C\int x^3\ dx = \frac{x^4}{4} + C

Definition of Definite Integral

∫abf(x)dx=lim⁡n→∞∑k=1nf(xk∗)⋅Δx\int^b_a f(x) dx = \lim\limits_{n \to \infty} \sum\limits^n_{k=1} f(x^*_k) \cdot \Delta x

  • Left point：xk∗=a+(k−1)Δxx_k^* = a + (k-1)\Delta x
  • Right point：xk∗=a+kΔxx_k^* = a + k\Delta x
  • Middle point：xk∗=a+(k−12)Δxx_k^* = a + \left(k - \frac{1}{2}\right)\Delta x

Definite Integral

∫abf(x)dx=lim⁡n→∞∑k=1nf(a+k⋅b−an)⋅b−an\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

∫abf(x)dx=lim⁡n→∞∑k=1nf(a+(k−1)⋅b−an)⋅b−an\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

∫abf(x)dx=lim⁡n→∞∑k=1nf(a+k⋅b−an)⋅b−an\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

∫abf(x)dx=lim⁡n→∞∑k=1nf(a+(k−12)⋅b−an)⋅b−an\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

∫abf(x)dx=lim⁡n→∞b−a2n[f(a)+2∑n−1k=1f(a+k⋅b−an+f(b))]\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

∫abf(x)dx=lim⁡n→∞∑k=1nf(a+(k−1)⋅b−an+f(a+k⋅b−an))2⋅b−an\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

  • ∫dx=x+C\boxed{\int dx = x + C}
  • ∫k dx=kx+C\boxed{\int k\ dx = kx + C}
  • ∫kf(x)dx=k∫f(x)dx\boxed{\int kf(x)dx = k \int f(x)dx}
  • ∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx\boxed{\int[f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx}
  • ∫xndx=1n+1xn+1+C\boxed{\int x^n dx = \frac1{n+1} x^{n+1} + C} & ∫1xdx=ln⁡∣x∣+C\boxed{\int \frac1x dx = \ln |x| + C}
  • ∫axdx=(1ln⁡a)ax+C\boxed{\int a^x dx = (\frac1{\ln a}) a^x + C}
  • ∫exdx=ex+C\int e^x dx = e^x + C
  • ∫sin⁡xdx=−cos⁡x+C\boxed{\int \sin x dx = - \cos x + C}
  • ∫cos⁡xdx=sin⁡x+C\boxed{\int \cos x dx = \sin x + C}
  • ∫sec⁡2x dx=−tan⁡x+C\boxed{\int \sec^2 x\ dx = -\tan x + C}
  • ∫csc⁡2x dx=−cot⁡x+C\int \csc^2 x\ dx = -\cot x + C
  • ∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec x \tan x \ dx = \sec x + C
  • ∫csc⁡xcot⁡x dx=−csc⁡x+C\int \csc x \cot x \ dx = - \csc x + C
  • ∫dx1−x2=arcsin⁡x+C\int \dfrac{dx}{\sqrt{1-x^2}} = \arcsin x + C
  • ∫dx1+x2=arctan⁡x+C\boxed{\int \dfrac{dx}{1+x^2}} = \arctan x + C
  • ∫dxa2−x2=arcsin⁡ac+C\int \dfrac{dx}{\sqrt{a^2-x^2}} = \arcsin \dfrac ac + C
  • ∫dxa2+x2=1aarctan⁡xa+C\int \dfrac{dx}{a^2+x^2} = \dfrac 1a \arctan \dfrac xa + C