# Integration of Algebraic and Trigonometric Functions

The following table contains integrated examples of basic algebraic and trigonometric formulas.
Ln means natural logarithm

 ꭍ dx x ꭍ xn.dx xn+1 / (n+1) ꭍ axn.dx a . xn+1 / (n+1) ꭍ (axn + b).dx= ꭍ axn.dx + ꭍ b.dx a.xn+1 / (n+1) + b.x ꭍ (ax + b)n.dx (ax+b)n+1 / a(n+1) ꭍ dx / (ax + b)= 1/a . ꭍ a.dx / (ax + b) 1/a . Ln(ax+b) ꭍ 1/x . dx Ln(x) ꭍ 1/(x + b)½ . dx 2(x+b)½ ꭍ 1/(ax + b)½ . dx 2(ax+b)½ / a ꭍ 1/(x² - a²) . dx -Acoth(x/a) / a or Ln[(x-a)/(x+a)] / 2a ꭍ 1/(a² - x²) . dx Atanh(x/a) / a or Ln[(a+x)/(a-x)] / 2a ꭍ 1/(a² + x²) . dx Atan(x/a) / a ꭍ (x² + a²)½ . dx ½x(x² + a²)½ + ½a² . Asinh(x/a) or ½x(x² + a²)½ + ½a² . Ln([x+(x² + a²)½] / a) ꭍ ƒ'(x)/ ƒ(x) . dx = Ln( ƒ(x)) Note: If the numerator = the differential of the denominator then the inverse of the denominator is the logₑ of the denominator. So multiply the equation by the differential of the denominator and 'Logₑ' the result d(u.v) / dx u.v = ꭍ u.dv/dx.dx +ꭍ v.du/dx.dx = ꭍ u.dv +ꭍ v.du ꭍ u.dv = u.v - ꭍ v.du ꭍ ax . dx ax . loga(e) ꭍ ex . dx ex ꭍ Sin(x) . dx -Cos(x) ꭍ Cos(x) . dx Sin(x) ꭍ Tan(x) . dx -Ln(Cos(x)), orLn(Sec(x)) ꭍ Cot(x) . dx Ln(Sin(x)) ꭍ Sec(x) . dx Ln(Tan(¼π + ½x)) ꭍ Cosec(x) . dx Ln(Tan(½x)) ꭍ Sinh(x) . dx Cosh(x) ꭍ Cosh(x) . dx Sinh(x) ꭍ Tanh(x) . dx Ln(Cosh(x)) ꭍ Coth(x) . dx Ln(Sinh(x)) ꭍ Sin(ax) . dx -Cos(ax) / a ꭍ Sin(ax + b) . dx -Cos(ax + b) / a ꭍ Cos(ax) . dx Sin(ax) / a ꭍ Cos(ax + b) . dx Sin(ax + b) / a ꭍ Tan(ax) . dx Ln(Sec(ax)) / a ꭍ Sinh(ax) . dx Cosh(ax) / a ꭍ Cosh(ax) . dx Sinh(ax) / a ꭍ Sin(x).Cos(x) . dx -¼Cos(2x) ꭍ Sec(x).Tan(x) . dx Sec(x) ꭍ Csc(x).Cot(x) . dx -Csc(x) ꭍ 1 / (a² - x²)½ . dx Asin(x/a), or-Acos(x/a) ꭍ 1 / (a² + x²) . dx Asec(x/a) / a, or-Acsc(x/a) / a ꭍ 1 / x(x² - a²)½ . dx Asec(x/a) / a, or-Acsc(x/a) / a ꭍ 1 / (x² + a²)½ . dx Asinh(x/a), orLn(x+(x²+a²)½ / a) ꭍ 1 / (x² - a²)½ . dx Acosh(x/a), orLn(x+(x²-a²)½ / a) ꭍ 1 / (a² - x²) . dx Atanh(x/a) / a, orLn((a+x)/(a-x)) / 2a ꭍ 1 / (x² - a²) . dx -Acoth(x/a) / a, orLn((a-x)/(a+x)) / 2a ꭍ 1 / x(a² - x²)½ . dx -Asech(x/a) / a, or-Ln((a + (a²-x²)½) / x) / a ꭍ 1 / x(a² + x²)½ . dx -Acsch(x/a) / a, or-Ln((a + (a²+x²)½) / x) / a ꭍ Sin²(x) . dx ½(x - ½.Sin(2x)) ꭍ Cos²(x) . dx ½(x + ½.Sin(2x)) ꭍ Tan²(x) . dx Tan(x) - x ꭍ Csc²(x) .dx -Cot(x) ꭍ Sec²(x) . dx Tan(x) ꭍ Cot²(x) . dx -(Cot(x) + x) ꭍ (x² - a²)½ .dx ½.x(x²-a²)½ - a².Acosh(x/a)/2, or½.x(x²-a²)½ - a²(logₑ((x+(x²-a²)½ / a) / 2 ꭍ (x² + a²)½ .dx ½.x(x²+a²)½ + a².Asinh(x/a)/2, or½x(x²+a²)½ + a²(logₑ((x+(x²+a²)½ / a) / 2 ꭍ (a² - x²)½ .dx ½.a².Asin(x/a) + ½.x(a² - x²)½ ꭍ Sin²(ax) ½x - ¼Sin(2ax)/a ꭍ x.Sin(ax).dx Sin(ax)/a² - x.Cos(ax)/a ꭍ x².Sin(ax) -x².Cos(ax)/a + 2.x.Sin(ax)/a² + 2Cos(ax)/a³ ꭍ x².Sin²(ax) x³/6 - ¼.x².Sin(2ax)/a - ¼x.Cos(2ax)/a² + ⅛Sin(2ax)/a³ ꭍ x³.Sin(ax) -x³.Cos(ax)/a + 3x².Sin(ax)/a² + 6.x.Cos(ax)/a³ - 6.Sin(ax)/a⁴ ꭍ Cos²(ax) ¼Sin(2ax)/a + ½x ꭍ x.Cos(ax).dx x.Sin(ax)/a + Cos(ax)/a² ꭍ x².Cos(ax) x².Sin(ax)/a + 2.x.Cos(ax)/a² - 2.Sin(ax)/a³ ꭍ x².Cos²(ax) ¼.x².Sin(2ax)/a + x³/6 + x.Cos(2ax) / 4a² - ⅛Sin(2ax)/a³ ꭍ x³.Cos(ax) x³.Sin(ax)/a + 3x².Cos(ax)/a² - 6.x.Sin(ax)/a³ - 6.Cos(ax)/a⁴ ꭍ Sin(x).Cos(x) -¼.Cos(2x)

## Worked Examples

The following table contains a number of examples worked through by CalQlata engineers from time to time.
The table may not yet be complete but will be eventually. We are adding new integral workings as we resolve them.

Note: there are a number of different ways to integrate these formulas, we have simply listed the methods we have used.

 Typical Integration by Substitution: Problem: ꭍ √[a + b.x²] . dx set: m = ꭍša; n = ꭍšb; x = m/n . Tan(θ)     {i.e. θ = Atan[x.n/m]} note: Sec²(θ) = 1+Tan²(θ) ꭍ (m² + n².x²)⁰ ™⁵ . dx      = ꭍ √[m² + n².m²/n² . Tan²[θ]] . dθ      = ꭍ √[m² + m² . Tan²[θ]] . dθ      = ꭍ √[m².(1 + Tan²[θ])] . dθ      = ꭍ √[m².Sec²[θ]] . dθ      = ꭍ m.Sec[θ] . dθ      = mꭍ Sec[θ] . dθ      = m . Ln(Tan[¼π + ½θ])     {see ꭍ Sec[x].dx above} substitute back: for x: m . Ln(Tan[¼π + ½{Atan[x.n/m]}]) for a & b: √a . Ln(Tan[¼π + ½{Atan[x.√b/√a]}]) ꭍ √[a + b.x²] . dx = √a . Ln(Tan(¼π + ½.Atan(x.√[b/a]))) ꭍ Sin²(x).dx Sin²(x) = Sin(x).Sin(x)      = ½(Cos(x-x) - Cos(x+x))      = ½(Cos(0) - Cos(2x))      = ½(1 - Cos(2x))      = ½ - ½Cos(2x) ꭍ Sin²(x) = ꭍ (½ - ½Cos(2x)).dx      = ꭍ ½.dx - ꭍ ½Cos(2x).dx      = ½ꭍ dx - ½ꭍ Cos(2x).dx      = ½.x - ½.Sin(2x)/2 ꭍ Sin²(x) = ½x - ¼Sin(2x) ꭍ Sin²(ax).dx Sin²(ax) = Sin(ax).Sin(ax)      = ½(Cos(ax-ax) - Cos(ax+ax))      = ½(Cos(0) - Cos(2ax))      = ½(1 - Cos(2ax))      = ½ - ½Cos(2ax) ꭍ Sin²(ax) = ꭍ (½ - ½Cos(ax)).dx      = ꭍ ½.dx - ꭍ ½Cos(2ax).dx      = ½ꭍ dx - ½ꭍ Cos(2ax).dx      = ½x - ½Sin(2ax)/2a ꭍ Sin²(ax) = ½x - ¼Sin(2ax)/a ꭍ x.Sin(ax).dx (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x; dv = Sin(ax); du = dx; v = -Cos(ax)/a ꭍ x.Sin(ax).dx = x.-Cos(ax)/a - ꭍ -Cos(ax)/a.dx      = -x.Cos(ax)/a + 1/aꭍ Cos(ax).dx      = -x.Cos(ax)/a + 1/a.Sin(ax)/a      = -x.Cos(ax)/a + Sin(ax)/a² ꭍ x.Sin(ax).dx = Sin(ax)/a² - x.Cos(ax)/a ꭍ x².Sin(ax) (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x²; dv = Sin(ax); du = 2x.dx; v = -Cos(ax)/a ꭍ x².Sin(ax) = x².-Cos(ax)/a - ꭍ -Cos(ax)/a . 2x.dx ꭍ x².Sin(ax) = -x².Cos(ax)/a + 2/aꭍ x.Cos(ax).dx ꭍ x.Cos(ax).dx u = x; dv = Cos(ax); du =dx; v = Sin(ax)/a ꭍ x.Cos(ax).dx = x . Sin(ax)/a - ꭍ Sin(ax)/a . dx      = x . Sin(ax)/a - 1/aꭍ Sin(ax).dx      = x . Sin(ax)/a - 1/a-Cos(ax)/a.dx      = x.Sin(ax)/a + Cos(ax)/a/a ꭍ x.Cos(ax).dx = x.Sin(ax)/a + Cos(ax)/a² ꭍ x².Sin(ax) = -x².Cos(ax)/a + 2/a . (x.Sin(ax)/a + Cos(ax)/a²)      = -x².Cos(ax)/a + (2/a . x.Sin(ax)/a + 2/a . Cos(ax)/a²)      = -x².Cos(ax)/a + (2x.Sin(ax)/a² + 2Cos(ax)/a³) ꭍ x².Sin(ax) = 2Cos(ax)/a³ + 2x.Sin(ax)/a² - x².Cos(ax)/a ꭍ x².Sin²(ax) Sin²(ax) = Sin(ax).Sin(ax)      = ½(Cos(ax-ax) - Cos(ax+ax))      = ½(Cos(0) - Cos(2ax))      = ½(1 - Cos(2ax)) Sin²(ax) = ½ - ½Cos(2ax) (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x²; dv = ½ - ½Cos(2ax); du = 2x.dx; v = ½x - ¼Sin(2ax)/a ꭍ x².Sin²(ax) = x².(½x - ¼.Sin(2ax)/a) - ꭍ (½x - ¼.Sin(2ax)/a) . 2x.dx      = ½x³ - ¼.x².Sin(2ax)/a - ꭍ (x² - ½.x.Sin(2ax)/a).dx      = ½x³ - ¼.x².Sin(2ax)/a - ꭍ x².dx + ꭍ ½.x.Sin(2ax)/a.dx      = ½x³ - ¼.x².Sin(2ax)/a - ꭍ x².dx + 1 / 2aꭍ x.Sin(2ax).dx      = ½x³ - ¼.x².Sin(2ax)/a - ⅓.x³ + 1 / 2aꭍ x.Sin(2ax).dx ꭍ x².Sin²(ax) = x³/6 - ¼.x².Sin(2ax)/a + 1 / 2aꭍ x.Sin(2ax).dx ꭍ x.Sin(2ax).dx u = x; dv = Sin(2ax); du = dx; v = -Cos(2ax)/2a ꭍ x.Sin(2ax).dx = -x.Cos(2ax) / 2a - ꭍ -Cos(2ax) / 2a . dx      = -x.Cos(2ax) / 2a + 1 / 2aꭍ Cos(2ax) . dx      = -x.Cos(2ax) / 2a + 1 / 2a.Sin(2ax) / 2a ꭍ x.Sin(2ax).dx = -x.Cos(2ax) / 2a + Sin(2ax) / 4a² ꭍ x².Sin²(ax) = x³/6 - ¼.x².Sin(2ax)/a + 1 / 2a . (-x.Cos(2ax) / 2a + Sin(2ax) / 4a²)      = x³/6 - ¼.x².Sin(2ax)/a + (-x.Cos(2ax) / 4a² + ⅛.Sin(2ax)/a³) ꭍ x².Sin²(ax) = x³/6 - ¼.x².Sin(2ax)/a - ¼x.Cos(2ax)/a² + ⅛.Sin(2ax)/a³ ꭍ x³.Sin(ax) (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x³; dv = Sin(ax); du = 3.x².dx; v = -Cos(ax)/a ꭍ x³.Sin(ax) = x³.-Cos(ax)/a - ꭍ -Cos(ax)/a . 3x².dx ꭍ x³.Sin(ax) = -x³.Cos(ax)/a + 3/aꭍ x².Cos(ax).dx ꭍ x².Cos(ax).dx u = x²; dv = Cos(ax); du =2x.dx; v = Sin(ax)/a ꭍ x².Cos(ax).dx = x².Sin(ax)/a - ꭍ Sin(ax)/a . 2x.dx      = x².Sin(ax)/a - 2/aꭍ Sin(ax) . x.dx ꭍ x².Cos(ax).dx = x².Sin(ax)/a - 2/aꭍ x.Sin(ax).dx ꭍ x.Sin(ax).dx u = x; dv = Sin(ax); du =dx; v = -Cos(ax)/a ꭍ x.Sin(ax).dx = x . -Cos(ax)/a - ꭍ -Cos(ax)/a . dx      = -x.Cos(ax)/a + 1/aꭍ Cos(ax).dx      = -x.Cos(ax)/a + Sin(ax)/a/a ꭍ x.Sin(ax).dx = -x.Cos(ax)/a + Sin(ax)/a² ꭍ x².Cos(ax).dx = x².Sin(ax)/a - 2/a . (-x.Cos(ax)/a + Sin(ax)/a²)      = x².Sin(ax)/a - (2/a.-x.Cos(ax)/a + 2/aSin(ax)/a²)      = x².Sin(ax)/a - (2.-x.Cos(ax)/a² + 2.Sin(ax)/a³) ꭍ x².Cos(ax).dx = x².Sin(ax)/a + 2.x.Cos(ax)/a² - 2.Sin(ax)/a³ ꭍ x³.Sin(ax) = -x³.Cos(ax)/a + 3/a . (x².Sin(ax)/a + 2.x.Cos(ax)/a² - 2.Sin(ax)/a³)      = -x³.Cos(ax)/a + (3/a . x².Sin(ax)/a + 3/a . 2.x.Cos(ax)/a² - 3/a . 2.Sin(ax)/a³)      = -x³.Cos(ax)/a + (3x².Sin(ax)/a² + 6.x.Cos(ax)/a³ - 6.Sin(ax)/a⁴) ꭍ x³.Sin(ax) = -x³.Cos(ax)/a + 3x².Sin(ax)/a² + 6.x.Cos(ax)/a³ - 6.Sin(ax)/a⁴ ꭍ Cos²(x).dx Cos²(x) = Cos(x).Cos(x)      = ½(Cos(x+x) + Cos(x-x))      = ½(Cos(2x) + Cos(0))      = ½(Cos(2x) + 1)      = ½Cos(2x) + ½ ꭍ Cos²(x) = ꭍ (½Cos(2x) + ½).dx      = ꭍ ½Cos(2x).dx + ꭍ ½.dx      = ½ꭍ Cos(2x).dx + ½ꭍ dx      = ½.Sin(2x)/2 + ½.x ꭍ Cos²(x) = ¼Sin(2x) + ½x ꭍ Cos²(ax).dx Cos²(ax) = Cos(ax).Cos(ax)      = ½(Cos(ax+ax) + Cos(ax-ax))      = ½(Cos(2ax) + Cos(0))      = ½(Cos(2ax) + 1)      = ½Cos(2ax) + ½ ꭍ Cos²(ax) = ꭍ (½Cos(ax) + ½).dx      = ꭍ ½Cos(2ax).dx + ꭍ ½.dx      = ½ꭍ Cos(2ax).dx + ½ꭍ dx      = ½Sin(2ax)/2a + ½x ꭍ Cos²(ax) = ¼Sin(2ax)/a + ½x ꭍ x.Cos(ax).dx (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x; dv = Cos(ax); du = dx; v = Sin(ax)/a ꭍ x.Cos(ax).dx = x.Sin(ax)/a - ꭍ Sin(ax)/a.dx      = x.Sin(ax)/a - 1/aꭍ Sin(ax).dx      = x.Sin(ax)/a - 1/a.-Cos(ax)/a      = x.Sin(ax)/a + Cos(ax)/a² ꭍ x.Cos(ax).dx = x.Sin(ax)/a + Cos(ax)/a² ꭍ x².Cos(ax) (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x²; dv = Cos(ax); du =2x.dx; v = Sin(ax)/a ꭍ x².Cos(ax) = x².Sin(ax)/a - ꭍ Sin(ax)/a . 2x.dx ꭍ x².Cos(ax) = x².Sin(ax)/a - 2/aꭍ x.Sin(ax).dx ꭍ x.Sin(ax).dx u = x; dv = Sin(ax); du =dx; v = -Cos(ax)/a      = x . -Cos(ax)/a - ꭍ -Cos(ax)/a . dx      = -x.Cos(ax)/a + 1/aꭍ Cos(ax).dx      = -x.Cos(ax)/a + 1/a.Sin(ax)/a ꭍ x.Sin(ax).dx = -x.Cos(ax)/a + Sin(ax)/a² ꭍ x².Cos(ax) = x².Sin(ax)/a - 2/a . (-x.Cos(ax)/a + Sin(ax)/a²)      = x².Sin(ax)/a - (2/a.-x.Cos(ax)/a + 2/a.Sin(ax)/a²)      = x².Sin(ax)/a - (2.-x.Cos(ax)/a² + 2.Sin(ax)/a³) ꭍ x².Cos(ax) = x².Sin(ax)/a + 2.x.Cos(ax)/a² - 2.Sin(ax)/a³ ꭍ x².Cos²(ax) Cos²(ax) = Cos(ax).Cos(ax)      = ½(Cos(ax+ax) + Cos(ax-ax))      = ½(Cos(2ax) + Cos(0))      = ½(Cos(2ax) + 1) Cos²(ax) = ½Cos(2ax) + ½ (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x²; dv = ½Cos(2ax) + ½; du = 2x.dx; v = ¼Sin(2ax)/a + ½x ꭍ x².Cos²(ax) = x².(¼.Sin(2ax)/a + ½x) - ꭍ (¼Sin(2ax)/a + ½x) . 2x.dx      = ¼.x².Sin(2ax)/a + ½x³ - ꭍ (½.x.Sin(2ax)/a + x²).dx      = ¼.x².Sin(2ax)/a + ½x³ - ꭍ ½.x.Sin(2ax)/a.dx - ꭍ x².dx      = ¼.x².Sin(2ax)/a + ½x³ - ꭍ x².dx - 1 / 2aꭍ x.Sin(2ax).dx      = ¼.x².Sin(2ax)/a + ½x³ - ⅓.x³ - 1 / 2aꭍ x.Sin(2ax).dx ꭍ x².Cos²(ax) = ¼.x².Sin(2ax)/a + x³/6 - 1 / 2aꭍ x.Sin(2ax).dx ꭍ x.Sin(2ax).dx u = x; dv = Sin(2ax); du = dx; v = -Cos(2ax) / 2a ꭍ x.Sin(2ax).dx = -x.Cos(2ax) / 2a - ꭍ -Cos(2ax) / 2a . dx      = -x.Cos(2ax) / 2a + 1 / 2aꭍ Cos(2ax) . dx      = -x.Cos(2ax) / 2a + 1 / 2a.Sin(2ax) / 2a . dx ꭍ x.Sin(2ax).dx = -x.Cos(2ax) / 2a + Sin(2ax) / 4a² ꭍ x².Cos²(ax) = ¼.x².Sin(2ax)/a + x³/6 - 1 / 2a . (-x.Cos(2ax) / 2a + Sin(2ax) / 4a²)      = ¼.x².Sin(2ax)/a + x³/6 - (-x.Cos(2ax) / 4a² + Sin(2ax) / 8a³) ꭍ x².Cos²(ax) = ¼.x².Sin(2ax)/a + x³/6 + x.Cos(2ax) / 4a² - ⅛.Sin(2ax)/a³ ꭍ x³.Cos(ax) (using integration by parts: ꭍ u.dv = uv - ꭍ v.du) u = x³; dv = Cos(ax); du =3x².dx; v = Sin(ax)/a ꭍ x³.Cos(ax) = x³.Sin(ax)/a - ꭍ Sin(ax)/a . 3x².dx ꭍ x³.Cos(ax) = x³.Sin(ax)/a - 3/aꭍ x².Sin(ax).dx ꭍ x².Sin(ax).dx u = x²; dv = Sin(ax); du =2x.dx; v = -Cos(ax)/a ꭍ x².Sin(ax).dx = x².-Cos(ax)/a - ꭍ -Cos(ax)/a . 2x.dx      = -x².Cos(ax)/a + 2/aꭍ Cos(ax) . x.dx ꭍ x².Sin(ax).dx = -x².Cos(ax)/a + 2/aꭍ x.Cos(ax).dx ꭍ x.Cos(ax).dx u = x; dv = Cos(ax); du =dx; v = Sin(ax)/a ꭍ x.Cos(ax).dx = x . Sin(ax)/a - ꭍ Sin(ax)/a . dx      = x . Sin(ax)/a - 1/aꭍ Sin(ax).dx      = x.Sin(ax)/a + Cos(ax)/a/a ꭍ x.Cos(ax).dx = x.Sin(ax)/a + Cos(ax)/a² ꭍ x².Sin(ax).dx = -x².Cos(ax)/a + 2/a . (x.Sin(ax)/a + Cos(ax)/a²)      = -x².Cos(ax)/a + (2/a . x.Sin(ax)/a + 2/a . Cos(ax)/a²)      = -x².Cos(ax)/a + (2.x.Sin(ax)/a² + 2.Cos(ax)/a³) ꭍ x².Sin(ax).dx = -x².Cos(ax)/a + 2.x.Sin(ax)/a² + 2.Cos(ax)/a³ ꭍ x³.Cos(ax) = x³.Sin(ax)/a - 3/a . (-x².Cos(ax)/a + 2.x.Sin(ax)/a² + 2.Cos(ax)/a³)      = x³.Sin(ax)/a - (3/a . -x².Cos(ax)/a + 3/a . 2.x.Sin(ax)/a² + 3/a . 2.Cos(ax)/a³)      = x³.Sin(ax)/a - (-3x².Cos(ax)/a² + 6.x.Sin(ax)/a³ + 6.Cos(ax)/a⁴) ꭍ x³.Cos(ax) = x³.Sin(ax)/a + 3x².Cos(ax)/a² - 6.x.Sin(ax)/a³ - 6.Cos(ax)/a⁴ ꭍ Sin(x).Cos(x).dx Sin(x).Cos(x) = ½(Sin(x+x) + Sin(x-x))      = ½(Sin(2x) + Sin(0))      = ½(Sin(2x) + 0)      = ½Sin(2x) ꭍ Sin(x).Cos(x) = ꭍ ½Sin(2x).dx      = ½ꭍ Sin(2x).dx      = ½.-Cos(2x)/2 ꭍ Sin(x).Cos(x) = -¼.Cos(2x) ꭍ Tan²(x).dx Tan²(x) = Sec²(x) - 1 ꭍ Tan²(x) = ꭍ (Sec²(x) - 1).dx      = ꭍ Sec²(x).dx - ꭍ dx ꭍ Tan²(x) = Tan(x) - x

Colour Coding is provided in the above table to assist with the flow/sequencing of some of the more complex calculations.