Cheat_Sheet

Theoretical Computer Science Cheat SheetCalculus Cont.Finite Calculus∫∫√dx62.x √ x 2 − a = 1 2 a arccos a|x| , a > 0, 63. dxx 2√ x 2 ± a = ∓ x2 ± a 2 Difference, shift operators:2 a 2 ,x∆f(x) =f(x +1)− f(x),∫xdx64. √x2 ± a = √ ∫ √x2x 2 ± a 2 ± a, 65.22 x 4 dx = ∓ (x2 + a 2 ) 3/2E f(x) =f(x +1).3a 2 x 3 ,∣1 ∣∣∣∣∫⎧⎪dx ⎨ √66.ax 2 + bx + c = b2 − 4ac ln 2ax + b − √ Fundamental Theorem:b 2 − 4ac2ax + b + √ b 2 − 4ac∣ , if f(x) =∆F (x) ⇔ ∑ f(x)δx = F (x)+C.b2 > 4ac,b∑ ∑b−1⎪ 2 ⎩ √ arctan √ 2ax + b , if f(x)δx = f(i).4ac − b2 4ac − b2 b2 < 4ac,ai=a1∣∫⎧⎪dx ⎨ √a ln ∣2ax + b +2 √ a √ Differences:ax 2 ∆(cu) =c∆u, ∆(u + v) =∆u +∆v,+ bx + c∣ , if a>0,67. √ax2 + bx + c = ⎪ 1 ⎩ √ arcsin √ −2ax − b∆(uv) =u∆v + E v∆u,−a b2 − 4ac ,if a0,if c 0,x 0 =1,x n 1=(x +1)···(x + |n|) , n < 0,x n+m = x m (x − m) n .Rising Factorial Powers:x n = x(x +1)···(x + n − 1), n > 0,x 0 =1,x n 1=(x − 1) ···(x −|n|) , n < 0,x n+m = x m (x + m) n .Conversion:x n =(−1) n (−x) n =(x − n +1) n=1/(x +1) −n ,x n =(−1) n (−x) n =(x + n − 1) n=1/(x − 1) −n ,n∑{ } nn∑{ } nx n = x k = (−1) n−k x k ,k kx n =x n =k=1n∑k=1n∑k=1k=1[ nk](−1) n−k x k ,[ nk]x k .