Exercise 3.3

MA111: Prepared by Dr.Archara Pacheenburawana 40
Exercise 4.4
1. Evaluate the definite integrals using the First Fundamental Theorem of Calculus.
(a)
(c)
(e)
(g)
(i)
(k)
(m)
(o)
(q)
(s)
(t)
∫ 3
−1
∫ 4
0
∫ 2
1
∫ 3
3
∫ π
π/2
∫ 9
1
∫ 9
8
∫ π/2
π/6
∫ 2
0
∫ 3
−2
∫ 2
0
x 5 dx
(b)
∫ 2
0
(2x−3)dx
∫
√ 4
xdx (d) ( √ x+3x)dx
∫
3
1
x dx (f) (x √ x+x −1/2 )dx
4 0
√
∫ π/2
x5 +2dx (h) 2sinxdx
secxtanxdx
(j)
0
0
∫ π
π/2
(2sinx−cosx)dx
∫
1
1
2x dx (l) (e x −e −x )dx
2 x dx (n)
(
x+ 2 )
sin 2 dx
x
(p)
|2x−3|dx
(r)
0
∫ 3
0
∫ √ 3
1
∫ 3π/4
0
(3e x −x 2 )dx
6
1+x 2 dx
|cosx|dx
{ −x, x ≥ 0
f(x)dx, where f(x) =
x 2 , x < 0
{ x
f(x)dx, where f(x) =
4 , 0 ≤ x < 1
x 5 , 1 ≤ x ≤ 2
2. Use the Second Fundamental Theorem of Calculus to find the derivative.
∫ x
∫ x √
(a) f(x) = (t 2 −3t+2)dt (b) g(x) = 1−2tdt
(c) g(y) =
(e) f(x) =
(g) h(x) =
(i) y =
∫ 1
0
∫ y
1−3x
2
∫ x 2
0
∫ 1/x
2
t 2 sintdt (d) F(x) =
(e −t2 +1)dt (f) f(x) =
arctantdt (h) y =
0
∫ 2
x
∫ −1
x
∫ √ x
u 3
∫ x 3
1+u du (j) y = 2
3
cos(t 2 )dt
cost
t
ln(t 2 +1)dt
dt
√ x
√
tsintdt