082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

454 Chapter 13 Integration
9 Byusingsuitable trigonometric identities find
∫
(a) sin 2 2t +cos 2 2t dt
∫
(b) sin2tcos2t dt
∫
1
(c)
sin2t dt
∫ cos2t
(d)
sin2t dt
∫ sin2t
(e)
cos2t dt
10 Byexpressing
2x 2 +x+2
x 3 +x
asits partialfractions, find
∫ 2x 2 +x+2
x 3 dx
+x
11 Evaluate the followingdefiniteintegrals:
∫ 1
(a) 2t 2 +t 3 dt
0
∫ 3 2
(b)
1 x − x 2 dx
∫ 1
(c) 7−t+7t 2 dt
0
∫ 2
(d) (z +1)(z +2)dz
0
∫ 4 √
(e) x dx
1
∫ −1 x −2
(f) dx
−2 x
12 Evaluate the followingdefiniteintegrals:
∫ 1
∫ 1
(a) e 3x+1 dx (b) e 2t −e t +1dt
0
−1
∫ 2
∫ 1
(c) (e z −1) 2 dz (d) e −2x +e 2x dx
1
0
∫ 0
(e) x+e x dx
−1
13 Evaluate the followingintegrals:
∫ π/2
(a) 2sin3t dt
0
∫ π
(b) sin2t −cos2t dt
π/2
∫ π/4
(c) tant+tdt
0
∫ ( ) ( )
π/4 t t
(d) sin +cos dt
0 2 2
∫ 0.1
(e) tan3t dt
0
14 Evaluate the following definiteintegrals:
∫ 2π/k
(a) sinkt dt
0
∫ 2π/k
(b) coskt dt
0
wherekis aconstant.
15 Evaluate the following definiteintegrals:
∫ 0.5
(a) cosec(2x +1)dx
0
∫ 0.1
(b) sec3t dt
0
∫ π/4
(c) tan(x + π)dx
−π/4
16 Evaluate the following definiteintegrals:
∫ 2 3
(a) √ dx
0 9−x 2
∫ 1 2
(b)
−1 9+x 2 dx
∫ 1 1
(c) √ dx
0 8−2x 2
∫ 3 1
(d)
1 10+4t 2 dt
17 (a) Calculate the area enclosed bythe curvey =x 3 ,
thexaxisandx = 2.
(b) Calculate the area enclosed bythe curvey =x 3 ,
theyaxisandy = 8.
18 Findthe total area between f (t) =t 2 −4 andthet
axisonthe followingintervals:
(a) [−4, −3] (b) [−3, −1]
(c) [0,3]
(d) [−3,3]
19 Calculate the area enclosed by the curve
y=x 2 −x−6andthexaxis.
20 Calculate the total area betweeny =x 2 −3x −4 and
thexaxisonthe followingintervals:
(a) [−2, −1] (b) [−2,1]
(c) [2,4] (d) [2,5]
21 Calculate the area enclosed by the curvey =x 3 and
theliney =x.