Integration of Algebraic and Trigonometric Functions

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 a^x . dx

a^x . log_a(e)

ꭍ e^x . dx

e^x

ꭍ Sin(x) . dx

-Cos(x)

ꭍ Cos(x) . dx

Sin(x)

ꭍ Tan(x) . dx

-Ln(Cos(x)), or
Ln(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), or
Ln(x+(x²+a²)^½ / a)

ꭍ 1 / (x² - a²)^½ . dx

Acosh(x/a), or
Ln(x+(x²-a²)^½ / a)

ꭍ 1 / (a² - x²) . dx

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

ꭍ 1 / (x² - a²) . dx

-Acoth(x/a) / a, or
Ln((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, or
½.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, or
½x(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)

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