• Clean, free, unlimited energy for allCLEAN, FREE, UNLIMITED ENERGY
  • Earth's renewable energyEARTH'S RENEWABLE ENERGY
  • Eliminate batteries, wind turbines and solar panelsELIMINATE BATTERIES, WIND TURBINES & SOLAR PANELS
  • 100% perfect medicines in minutesPERFECT MEDICINES IN MINUTES
  • Eliminate surface skin-frictionELIMINATE SKIN FRICTION
Clean, free, unlimited energy1 Earth's renewable energy2 Eliminate batteries, wind turbines and solar panels3 100% perfect medicines in minutes4 Eliminate surface skin-friction5
Useful Stuff Algebra Trig Functions Calculus Max-Min Differentiation Rules Differentiation Trig Differentiation Logs Integration Methods Standard Integrals Stiffness & Capacity Mohr's Circle Earth's Atmosphere Earth's Properties Stars & Black Holes Stars & The Gas Planets Laws of Motion Solar System Orbits Planetary Spin Core Pressure Earth's Magnetic Field Dark Matter? The Big Bang Rydberg Atom Planck Atom Classical Atom Newton Atom The Atom Newton's 'G' Coulomb's 'k' The Neutron E=mc² Gravity is Magnetism Relativity is Dead Quantum Theory is Dead Artificial Satellites Brakes and Tyres Vehicle Impacts Speeding vs Safety Surface Finish Pressure Classes Hardness Conversion Energy Electro-Magnetic Spectra Thermodynamics Steam (properties) Heat Work Energy Power Constants

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)), or
Ln(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), or
Ln(x+(x²+a²)½ / a)

∫1 / (x² – a²)½ . dx

Acosh(x/a), or
Ln(x+(x²–a²)½ / a)

∫1 / (a² – x²) . dx

Atanh(x/a) / a, or
Ln((a+x)/(a–x)) / 2a

∫1 / (x² – a²) . dx

–Acoth(x/a) / a, or
Ln((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, or
½.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, or
½x(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 + .m²/ . 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.

Further Reading

You will find further reading on this subject in reference publications(19)

      Go to our store
CalQlata™ Copyright ©2011-2019 CalQlata info@calqlata.com Site Map Terms of website use Our Store