1. ∫A dx or ∫A+B dx
1.1. simple integration
1.1.1. ∫Xⁿ dx
1.1.1.1. ⅟n+1 Xⁿ⁺¹ + c
1.1.2. ∫eˣ dx
1.1.2.1. eˣ + c
1.2. substitution
1.2.1. ∫X¹dx
1.2.1.1. u=x
1.2.1.2. du=1.dx
1.2.1.3. ∫u.du
1.2.1.3.1. 1/2 u² + c
1.3. exception (by parts)
1.3.1. lnx -> 1.lnx
1.3.1.1. dy/dx = 1/x
1.3.2. tan⁻¹x -> 1.tan⁻¹x
1.3.2.1. dy/dx = 1/(1+x²)
1.3.3. sin⁻¹x -> 1.sin⁻¹x
1.3.3.1. dy/dx = 1/√(1-x²)
2. ∫A.B dx
2.1. by parts
2.1.1. ∫uv' dx = uv - ∫u'v dx
2.1.2. order in ILATE
2.1.2.1. inverse trigonometric
2.1.2.2. logarithmic
2.1.2.3. algebraic
2.1.2.4. trigonometric
2.1.2.5. exponential
2.2. trigonometry
2.2.1. double angle formula
2.2.1.1. cos2x
2.2.1.2. sin2x
2.2.1.3. tan2x
3. ∫A/(X+B) dx
3.1. partial fractions
3.1.1. 1/(x+c)(x+d)
3.1.1.1. A/(x+c) + B/(x+d)
3.1.2. 1/(x²+d)(x+e)
3.1.2.1. Ax+B/(x²+d)+ C/(x+e)
3.1.3. 1/(x+d)²(x+e)
3.1.3.1. A/(x+d) + B/(x+d)² + C/(x+e)
3.2. notes
3.2.1. denominator = xⁿ⁼¹
3.2.1.1. ln x
3.2.2. denominator = xⁿ>¹
3.2.2.1. single: power rule
3.2.2.2. multiple: tan⁻¹x / sin⁻¹x / substitution
4. ∫A+B/X dx
4.1. seperate
4.1.1. use power rule/simple integration
4.1.2. trigonometric (identity/double ange)
5. trigonometry
5.1. Trigonometry to the power of 1
5.1.1. with x
5.1.1.1. sinx -> -cosx
5.1.1.2. cosx -> sinx
5.1.1.3. sec²x -> tanx
5.1.2. with nx
5.1.2.1. sinax -> -1/a cosax
5.1.2.2. cosax -> 1/a sinax
5.1.2.3. sec²ax -> 1/a tanax
5.2. Trigonometry to power of more than 1
5.2.1. Use identity
5.2.1.1. cos²x+sin²x= 1
5.2.1.2. sin²x= (1-cos²x)
5.2.1.3. cos²x= (
5.2.1.4. sec²a = 1 + tan²a
5.2.1.5. seca = 1/cosa
5.2.1.6. coseca = 1/sina
5.2.1.7. cota = 1/tana
5.2.2. Double Angle Formula
5.2.2.1. cos2a
5.2.2.1.1. cos²a - sin²a
5.2.2.1.2. 2cos²a - 1
5.2.2.1.3. 1 - 2sin²a
5.2.2.2. sin2a
5.2.2.2.1. 2sinacosa
5.2.2.3. tan2a
5.2.2.3.1. 2tana/(1-tan²a)
5.3. Trigonometry with function inside it
5.3.1. u' = sin (ax+b)
5.3.1.1. u = -1/a cos (ax+b)
5.3.2. u' = cos (ax+b)
5.3.2.1. 1/a sin (ax+b)
5.3.3. u' = sec² (ax+b)
5.3.3.1. u= 1/a tan(ax+b)