Integration

3. Area of a Square
Area of a Rectangle
Area of a circle
Area of a Triangle
Area of a Trapezium
ℝ - real numbers, 𝜖 – belongs to

4.  Function 𝐹(𝑥) is an antiderivative of
𝑓(𝑥) if
𝐹′ 𝑥 = 𝑓 𝑥
 The approach of finding an
antiderivative is called
antidifferentiation or integration.
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝑐
𝑐 is a constant, 𝑑𝑥 is the variable of
integration and 𝑓(𝑥) is Integrand
The above equation is read as
“integral of 𝑓(𝑥) with respect to 𝑥 is
𝐹 𝑥 ”.

5.  𝑘, 𝑐 𝜖 ℝ
 𝑥𝑛𝑑𝑥 =
𝑥𝑛+1
𝑛+1
+ 𝑐 (𝑛 ≠ −1)
 𝑐𝑑𝑥 = 𝑐𝑥 + 𝑘

1
𝑥
𝑑𝑥 = ln 𝑥 + 𝑐
 𝑐𝑓 𝑥 𝑑𝑥 = 𝑐 𝑓 𝑥 𝑑𝑥, where 𝑐 𝜖 ℝ
 [𝑓 𝑥 ± 𝑔 𝑥 ]𝑑𝑥 =
𝑓 𝑥 𝑑𝑥 ± 𝑔 𝑥 𝑑𝑥
 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝑐

7.  𝑎, 𝑏, 𝑐 𝜖 ℝ. (ℝ - real numbers, 𝜖 –
belongs to)
 Given 𝑓(𝑥)𝑑𝑥 = 𝐹 𝑥
 𝑓(𝑎𝑥 + 𝑏)𝑑𝑥 =
1
𝑎
𝐹 𝑎𝑥 + 𝑏 + 𝑐
 (𝑎𝑥 + 𝑏)𝑛
𝑑𝑥 =
(𝑎𝑥+𝑏)𝑛+1
𝑎(𝑛+1)
+ 𝑐

1
𝑎𝑥+𝑏
𝑑𝑥 =
1
𝑎
ln 𝑎𝑥 + 𝑏 + 𝑐
 𝑒𝑎𝑥+𝑏𝑑𝑥 =
1
𝑎
𝑒𝑎𝑥+𝑏 + 𝑐
 Constant 𝑐 is obatined by
intial/boundary condition.

8.  If velocity and acceleration have the same sign, the body is accelerating.
 If velocity and acceleration are in the opposite sign, the body is decelerating.
Displacement
s(t)
Acceleration
a(t)
Velocity
v(t)
𝑣 𝑡 𝑑𝑡 = 𝑠 𝑡 + 𝑐 𝑎 𝑡 𝑑𝑡 = 𝑣 𝑡 + 𝑐

10.  𝑎, 𝑏, 𝑐 𝜖 ℝ
 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 - integral from 𝑎 to 𝑏 of 𝑓
with respect to 𝑥.
 𝑎
𝑎
𝑓 𝑥 𝑑𝑥 = 0
 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = − 𝑏
𝑎
𝑓 𝑥 𝑑𝑥
 𝑎
𝑏
𝑘𝑓 𝑥 𝑑𝑥 = 𝑘 𝑎
𝑏
𝑓 𝑥 𝑑𝑥
 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑎
𝑐
𝑓 𝑥 𝑑𝑥 + 𝑐
𝑏
𝑓 𝑥 𝑑𝑥
(𝑎 < 𝑥 < 𝑏)
 𝑎
𝑏
[𝑓 𝑥 ± 𝑔 𝑥 ]𝑑𝑥 =
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 ± 𝑎
𝑏
𝑔 𝑥 𝑑𝑥

11. First and Second Derivatives

12.  If 𝐹′ 𝑥 = 𝑓 𝑥 i.e 𝐹 𝑥 is an
antiderivative of 𝑓 𝑥 and 𝑓 𝑥 is
continuous in the interval 𝑎 < 𝑥 < 𝑏
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 = 𝐹 𝑏 − 𝐹(𝑎)

13. Optimization and Kinematics

14.  Area under a curve
 If 𝑓 𝑥 and 𝑔 𝑥 are continuous on
𝑎 < 𝑥 < 𝑏 and 𝑓 𝑥 > 𝑔(𝑥)
 The area between the curves 𝑓 𝑥 and
𝑔 𝑥 is given by
𝑎
𝑏
[𝑓 𝑥 − 𝑔(𝑥)]𝑑𝑥