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Integration of Algebraic and Trigonometric Functions

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

 ∫dx x ∫xn.dx xn+1 / (n+1) ∫axn.dx a . xn+1 / (n+1) ∫(axn + b).dx= ∫axn.dx + ∫b.dx a.xn+1 / (n+1) + b.x ∫(ax + b)n.dx (ax+b)n+1 / a(n+1) ∫dx / (ax + b)= 1/a . ∫a.dx / (ax + b) 1/a . Ln(ax+b) ∫1/x . dx Ln(x) ∫1/(x + b)½ . dx 2(x+b)½ ∫1/(ax + b)½ . dx 2(ax+b)½ / a ∫1/(x² - a²) . dx -Acoth(x/a) / a or Ln[(x-a)/(x+a)] / 2a ∫1/(a² - x²) . dx Atanh(x/a) / a or Ln[(a+x)/(a-x)] / 2a ∫1/(a² + x²) . dx Atan(x/a) / a ∫(x² + a²)½ . dx ½x(x² + a²)½ + ½a² . Asinh(x/a) or ½x(x² + a²)½ + ½a² . Ln([x+(x² + a²)½] / a) ∫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 = ∫u.dv/dx.dx +∫v.du/dx.dx = ∫u.dv +∫v.du ∫u.dv = u.v - ∫v.du ∫ax . dx ax . loga(e) ∫ex . dx ex ∫Sin(x) . dx –Cos(x) ∫Cos(x) . dx Sin(x) ∫Tan(x) . dx –Ln(Cos(x)), orLn(Sec(x)) ∫Cot(x) . dx Ln(Sin(x)) ∫Sec(x) . dx Ln(Tan(¼π + ½x)) ∫Cosec(x) . dx Ln(Tan(½x)) ∫Sinh(x) . dx Cosh(x) ∫Cosh(x) . dx Sinh(x) ∫Tanh(x) . dx Ln(Cosh(x)) ∫Coth(x) . dx Ln(Sinh(x)) ∫Sin(ax) . dx –Cos(ax) / a ∫Sin(ax + b) . dx –Cos(ax + b) / a ∫Cos(ax) . dx Sin(ax) / a ∫Cos(ax + b) . dx Sin(ax + b) / a ∫Tan(ax) . dx Ln(Sec(ax)) / a ∫Sinh(ax) . dx Cosh(ax) / a ∫Cosh(ax) . dx Sinh(ax) / a ∫Sin(x).Cos(x) . dx -¼Cos(2x) ∫Sec(x).Tan(x) . dx Sec(x) ∫Csc(x).Cot(x) . dx –Csc(x) ∫1 / (a² – x²)½ . dx Asin(x/a), or–Acos(x/a) ∫1 / (a² + x²) . dx Asec(x/a) / a, or–Acsc(x/a) / a ∫1 / x(x² – a²)½ . dx Asec(x/a) / a, or–Acsc(x/a) / a ∫1 / (x² + a²)½ . dx Asinh(x/a), orLn(x+(x²+a²)½ / a) ∫1 / (x² – a²)½ . dx Acosh(x/a), orLn(x+(x²–a²)½ / a) ∫1 / (a² – x²) . dx Atanh(x/a) / a, orLn((a+x)/(a–x)) / 2a ∫1 / (x² – a²) . dx –Acoth(x/a) / a, orLn((a–x)/(a+x)) / 2a ∫1 / x(a² – x²)½ . dx –Asech(x/a) / a, or–Ln((a + (a²–x²)½) / x) / a ∫1 / x(a² + x²)½ . dx –Acsch(x/a) / a, or–Ln((a + (a²+x²)½) / x) / a ∫Sin²(x) . dx ½(x – ½.Sin(2x)) ∫Cos²(x) . dx ½(x + ½.Sin(2x)) ∫Tan²(x) . dx Tan(x) – x ∫Csc²(x) .dx –Cot(x) ∫Sec²(x) . dx Tan(x) ∫Cot²(x) . dx –(Cot(x) + x) ∫(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 ∫(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 ∫(a² – x²)½ .dx ½.a².Asin(x/a) + ½.x(a² – x²)½ ∫Sin²(ax) ½x – ¼Sin(2ax)/a ∫x.Sin(ax).dx Sin(ax)/a² – x.Cos(ax)/a ∫x².Sin(ax) -x².Cos(ax)/a + 2.x.Sin(ax)/a² + 2Cos(ax)/a³ ∫x².Sin²(ax) x³/6 – ¼.x².Sin(2ax)/a – ¼x.Cos(2ax)/a² + ⅛Sin(2ax)/a³ ∫x³.Sin(ax) -x³.Cos(ax)/a + 3x².Sin(ax)/a² + 6.x.Cos(ax)/a³ – 6.Sin(ax)/a⁴ ∫Cos²(ax) ¼Sin(2ax)/a + ½x ∫x.Cos(ax).dx x.Sin(ax)/a + Cos(ax)/a² ∫x².Cos(ax) x².Sin(ax)/a + 2.x.Cos(ax)/a² – 2.Sin(ax)/a³ ∫x².Cos²(ax) ¼.x².Sin(2ax)/a + x³/6 + x.Cos(2ax) / 4a² – ⅛Sin(2ax)/a³ ∫x³.Cos(ax) x³.Sin(ax)/a + 3x².Cos(ax)/a² – 6.x.Sin(ax)/a³ – 6.Cos(ax)/a⁴ ∫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: ∫(a + b.x²)⁰˙⁵ . dx set: m = √a; n = √b; x = m/n . Tan(θ)     {i.e. θ = Atan[x.n/m]} note: Sec²(θ) = 1+Tan²(θ) ∫(m² + n².x²)⁰˙⁵ . dx      = ∫(m² + n².m²/n² . Tan²[θ])⁰˙⁵ . dθ      = ∫(m² + m² . Tan²[θ])⁰˙⁵ . dθ      = ∫(m².(1 + Tan²[θ]))⁰˙⁵ . dθ      = ∫(m².Sec²[θ])⁰˙⁵ . dθ      = ∫m.Sec[θ] . dθ      = m∫Sec[θ] . dθ      = m . Ln(Tan[¼π + ½θ])     {see ∫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]}]) ∫(a + b.x²)⁰˙⁵ . dx = √a . Ln(Tan[¼π + ½.Atan[x.√(b/a)]]) ∫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) ∫Sin²(x) = ∫(½ – ½Cos(2x)).dx      = ∫½.dx – ∫½Cos(2x).dx      = ½∫dx – ½∫Cos(2x).dx      = ½.x – ½.Sin(2x)/2 ∫Sin²(x) = ½x – ¼Sin(2x) ∫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) ∫Sin²(ax) = ∫(½ – ½Cos(ax)).dx      = ∫½.dx – ∫½Cos(2ax).dx      = ½∫dx – ½∫Cos(2ax).dx      = ½x – ½Sin(2ax)/2a ∫Sin²(ax) = ½x – ¼Sin(2ax)/a ∫x.Sin(ax).dx (using integration by parts: ∫u.dv = uv - ∫v.du) u = x; dv = Sin(ax); du = dx; v = -Cos(ax)/a ∫x.Sin(ax).dx = x.-Cos(ax)/a – ∫-Cos(ax)/a.dx      = -x.Cos(ax)/a + 1/a∫Cos(ax).dx      = -x.Cos(ax)/a + 1/a.Sin(ax)/a      = -x.Cos(ax)/a + Sin(ax)/a² ∫x.Sin(ax).dx = Sin(ax)/a² – x.Cos(ax)/a ∫x².Sin(ax) (using integration by parts: ∫u.dv = uv - ∫v.du) u = x²; dv = Sin(ax); du = 2x.dx; v = -Cos(ax)/a ∫x².Sin(ax) = x².-Cos(ax)/a - ∫-Cos(ax)/a . 2x.dx ∫x².Sin(ax) = -x².Cos(ax)/a + 2/a∫x.Cos(ax).dx ∫x.Cos(ax).dx u = x; dv = Cos(ax); du =dx; v = Sin(ax)/a ∫x.Cos(ax).dx = x . Sin(ax)/a – ∫Sin(ax)/a . dx      = x . Sin(ax)/a – 1/a∫Sin(ax).dx      = x . Sin(ax)/a – 1/a-Cos(ax)/a.dx      = x.Sin(ax)/a + Cos(ax)/a/a ∫x.Cos(ax).dx = x.Sin(ax)/a + Cos(ax)/a² ∫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) ∫x².Sin(ax) = 2Cos(ax)/a³ + 2x.Sin(ax)/a² – x².Cos(ax)/a ∫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: ∫u.dv = uv - ∫v.du) u = x²; dv = ½ – ½Cos(2ax); du = 2x.dx; v = ½x – ¼Sin(2ax)/a ∫x².Sin²(ax) = x².(½x – ¼.Sin(2ax)/a) – ∫(½x – ¼.Sin(2ax)/a) . 2x.dx      = ½x³ – ¼.x².Sin(2ax)/a – ∫(x² – ½.x.Sin(2ax)/a).dx      = ½x³ – ¼.x².Sin(2ax)/a – ∫x².dx + ∫½.x.Sin(2ax)/a.dx      = ½x³ – ¼.x².Sin(2ax)/a – ∫x².dx + 1 / 2a∫x.Sin(2ax).dx      = ½x³ – ¼.x².Sin(2ax)/a – ⅓x³ + 1 / 2a∫x.Sin(2ax).dx ∫x².Sin²(ax) = x³/6 – ¼.x².Sin(2ax)/a + 1 / 2a∫x.Sin(2ax).dx ∫x.Sin(2ax).dx u = x; dv = Sin(2ax); du = dx; v = -Cos(2ax)/2a ∫x.Sin(2ax).dx = -x.Cos(2ax) / 2a – ∫-Cos(2ax) / 2a . dx      = -x.Cos(2ax) / 2a + 1 / 2a∫Cos(2ax) . dx      = -x.Cos(2ax) / 2a + 1 / 2a.Sin(2ax) / 2a ∫x.Sin(2ax).dx = -x.Cos(2ax) / 2a + Sin(2ax) / 4a² ∫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 + ⅛Sin(2ax)/a³) ∫x².Sin²(ax) = x³/6 – ¼.x².Sin(2ax)/a – ¼x.Cos(2ax)/a² + ⅛Sin(2ax)/a³ ∫x³.Sin(ax) (using integration by parts: ∫u.dv = uv - ∫v.du) u = x³; dv = Sin(ax); du = 3.x².dx; v = -Cos(ax)/a ∫x³.Sin(ax) = x³.-Cos(ax)/a – ∫-Cos(ax)/a . 3x².dx ∫x³.Sin(ax) = -x³.Cos(ax)/a + 3/a∫x².Cos(ax).dx ∫x².Cos(ax).dx u = x²; dv = Cos(ax); du =2x.dx; v = Sin(ax)/a ∫x².Cos(ax).dx = x².Sin(ax)/a – ∫Sin(ax)/a . 2x.dx      = x².Sin(ax)/a – 2/a∫Sin(ax) . x.dx ∫x².Cos(ax).dx = x².Sin(ax)/a – 2/a∫x.Sin(ax).dx ∫x.Sin(ax).dx u = x; dv = Sin(ax); du =dx; v = -Cos(ax)/a ∫x.Sin(ax).dx = x . -Cos(ax)/a – ∫-Cos(ax)/a . dx      = -x.Cos(ax)/a + 1/a∫Cos(ax).dx      = -x.Cos(ax)/a + Sin(ax)/a/a ∫x.Sin(ax).dx = -x.Cos(ax)/a + Sin(ax)/a² ∫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) ∫x².Cos(ax).dx = x².Sin(ax)/a + 2.x.Cos(ax)/a² – 2.Sin(ax)/a³ ∫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) ∫x³.Sin(ax) = -x³.Cos(ax)/a + 3x².Sin(ax)/a² + 6.x.Cos(ax)/a³ – 6.Sin(ax)/a⁴ ∫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 ∫Cos²(x) = ∫(½Cos(2x) + ½).dx      = ∫½Cos(2x).dx + ∫½.dx      = ½∫Cos(2x).dx + ½∫dx      = ½.Sin(2x)/2 + ½.x ∫Cos²(x) = ¼Sin(2x) + ½x ∫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 ∫Cos²(ax) = ∫(½Cos(ax) + ½).dx      = ∫½Cos(2ax).dx + ∫½.dx      = ½∫Cos(2ax).dx + ½∫dx      = ½Sin(2ax)/2a + ½x ∫Cos²(ax) = ¼Sin(2ax)/a + ½x ∫x.Cos(ax).dx (using integration by parts: ∫u.dv = uv - ∫v.du) u = x; dv = Cos(ax); du = dx; v = Sin(ax)/a ∫x.Cos(ax).dx = x.Sin(ax)/a – ∫Sin(ax)/a.dx      = x.Sin(ax)/a – 1/a∫Sin(ax).dx      = x.Sin(ax)/a – 1/a.-Cos(ax)/a      = x.Sin(ax)/a + Cos(ax)/a² ∫x.Cos(ax).dx = x.Sin(ax)/a + Cos(ax)/a² ∫x².Cos(ax) (using integration by parts: ∫u.dv = uv - ∫v.du) u = x²; dv = Cos(ax); du =2x.dx; v = Sin(ax)/a ∫x².Cos(ax) = x².Sin(ax)/a – ∫Sin(ax)/a . 2x.dx ∫x².Cos(ax) = x².Sin(ax)/a – 2/a∫x.Sin(ax).dx ∫x.Sin(ax).dx u = x; dv = Sin(ax); du =dx; v = -Cos(ax)/a      = x . -Cos(ax)/a – ∫-Cos(ax)/a . dx      = -x.Cos(ax)/a + 1/a∫Cos(ax).dx      = -x.Cos(ax)/a + 1/a.Sin(ax)/a ∫x.Sin(ax).dx = -x.Cos(ax)/a + Sin(ax)/a² ∫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) ∫x².Cos(ax) = x².Sin(ax)/a + 2.x.Cos(ax)/a² – 2.Sin(ax)/a³ ∫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: ∫u.dv = uv - ∫v.du) u = x²; dv = ½Cos(2ax) + ½; du = 2x.dx; v = ¼Sin(2ax)/a + ½x ∫x².Cos²(ax) = x².(¼.Sin(2ax)/a + ½x) – ∫(¼Sin(2ax)/a + ½x) . 2x.dx      = ¼.x².Sin(2ax)/a + ½x³ – ∫(½.x.Sin(2ax)/a + x²).dx      = ¼.x².Sin(2ax)/a + ½x³ – ∫½.x.Sin(2ax)/a.dx – ∫x².dx      = ¼.x².Sin(2ax)/a + ½x³ – ∫x².dx – 1 / 2a∫x.Sin(2ax).dx      = ¼.x².Sin(2ax)/a + ½x³ – ⅓x³ – 1 / 2a∫x.Sin(2ax).dx ∫x².Cos²(ax) = ¼.x².Sin(2ax)/a + x³/6 – 1 / 2a∫x.Sin(2ax).dx ∫x.Sin(2ax).dx u = x; dv = Sin(2ax); du = dx; v = -Cos(2ax) / 2a ∫x.Sin(2ax).dx = -x.Cos(2ax) / 2a - ∫-Cos(2ax) / 2a . dx      = -x.Cos(2ax) / 2a + 1 / 2a∫Cos(2ax) . dx      = -x.Cos(2ax) / 2a + 1 / 2a.Sin(2ax) / 2a . dx ∫x.Sin(2ax).dx = -x.Cos(2ax) / 2a + Sin(2ax) / 4a² ∫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) ∫x².Cos²(ax) = ¼.x².Sin(2ax)/a + x³/6 + x.Cos(2ax) / 4a² – ⅛Sin(2ax)/a³ ∫x³.Cos(ax) (using integration by parts: ∫u.dv = uv - ∫v.du) u = x³; dv = Cos(ax); du =3x².dx; v = Sin(ax)/a ∫x³.Cos(ax) = x³.Sin(ax)/a – ∫Sin(ax)/a . 3x².dx ∫x³.Cos(ax) = x³.Sin(ax)/a – 3/a∫x².Sin(ax).dx ∫x².Sin(ax).dx u = x²; dv = Sin(ax); du =2x.dx; v = -Cos(ax)/a ∫x².Sin(ax).dx = x².-Cos(ax)/a – ∫-Cos(ax)/a . 2x.dx      = -x².Cos(ax)/a + 2/a∫Cos(ax) . x.dx ∫x².Sin(ax).dx = -x².Cos(ax)/a + 2/a∫x.Cos(ax).dx ∫x.Cos(ax).dx u = x; dv = Cos(ax); du =dx; v = Sin(ax)/a ∫x.Cos(ax).dx = x . Sin(ax)/a – ∫Sin(ax)/a . dx      = x . Sin(ax)/a – 1/a∫Sin(ax).dx      = x.Sin(ax)/a + Cos(ax)/a/a ∫x.Cos(ax).dx = x.Sin(ax)/a + Cos(ax)/a² ∫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) ∫x².Sin(ax).dx = -x².Cos(ax)/a + 2.x.Sin(ax)/a² + 2.Cos(ax)/a³ ∫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) ∫x³.Cos(ax) = x³.Sin(ax)/a + 3x².Cos(ax)/a² – 6.x.Sin(ax)/a³ – 6.Cos(ax)/a⁴ ∫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) ∫Sin(x).Cos(x) = ∫½Sin(2x).dx      = ½∫Sin(2x).dx      = ½.-Cos(2x)/2 ∫Sin(x).Cos(x) = -¼.Cos(2x) ∫Tan²(x).dx Tan²(x) = Sec²(x) – 1 ∫Tan²(x) = ∫(Sec²(x) – 1).dx      = ∫Sec²(x).dx – ∫dx ∫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.