1. Find the value of the following integral.
(1) ∫ 𝑥√𝑥 + 4𝑑𝑥
(2) ∫ 𝑥 2 √𝑥 − 1𝑑𝑥
2. Find the value of the following integral.
(1) ∫
(2) ∫
𝑥
√1+𝑥2 √1+√1+𝑥2
𝑑𝑥
5
√√𝑥−1+(𝑥−1)4
𝑑𝑥
1
3. Find ∫−1 ( 4
1
√|𝑥|3
+ sin 𝑥 5 ) 𝑑𝑥 = ?
1
4. Find ∫ 1+sin(𝑎𝜃) 𝑑𝜃 = ? 𝑎 > 0
𝑏
5. There are two numbers a and b, such that ∫𝑎 (4 − 𝑥 2 )𝑑𝑥 has a maximum value.
Find the value of b - a = ?
6. Prove the following equation.
𝑎
𝑎
(1) ∫0 𝑓(𝑥)𝑑𝑥 = ∫0 𝑓(𝑎 − 𝑥)𝑑𝑥
𝜋
sin𝑛 𝑥
𝜋
sin𝑛 𝑥
𝜋
cos𝑛 𝑥
(2) ∫02 sin𝑛 𝑥+cos𝑛 𝑥 𝑑𝑥 = ∫02 sin𝑛 𝑥+cos𝑛 𝑥 𝑑𝑥
𝜋
(3) ∫02 sin𝑛 𝑥+cos𝑛 𝑥 𝑑𝑥 = 4
𝜋 −𝜋
7. What is the area of the region enclosed by y = cos 𝑥 , 𝑦 = cos 𝑥 , x ∈ [ 6 , 6 ] ?