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

Review exercises 13 453
6
7
1
6
4
3
8 (a) 1.0839 (b) 0.6468
(c) 0 (d) 2.3504
9 4.7756
10 0.4142
11 39
12 2.3026
13 (a) −3.6202 (b) 3.1945
(c) 0 (d) 5.4158
REVIEWEXERCISES13
1 Find the following integrals:
∫
∫ 2
(a) 3x 2 +xdx (b)
t +2t+2dt
∫
∫ √t 1
(c) (1 +z)(1 −z)dz (d) − √t dt
(e)
∫
2 Given
x 2/3 +4x 3 dx
dy
dx =x2 +sinx+cos2x+1
findan expression fory(x).
3 Find the following integrals:
∫
(a) 3e x + 3 e x dx
∫
(b) (1+e x )(1−e −x )dx
∫
(c) 2e 4t +1 dt
∫
(d) e x (1+e x )dx
∫
(e) 4e −t −e −2t dt
4 Find the integrals
∫
(a) sin2x +cos2x dx
∫
(b) 2sint−costdt
∫ ( )
t
(c) 4tan dt
2
∫
(d) sin(π −z) +cos(π −2z)dz
∫
(e) tan(t + π)dt
∫ ( )
t
(f) 2sin3t+2sin dt
3
5 Find the following integrals:
∫
(a) cosec(3t + π)dt
∫ ( )
x
(b) sec
2 +1 dx
∫ ( )
π +t
(c) cot dt
2
∫ ( )
y
(d) 3cosec
3 −2 dy
∫ 1
(e) cot(π −2z)dz
2
∫ 2
(f) sec(2t − π)dt
3
6 Find the following integrals:
∫
∫
4
(a) √ dv (b) 1−v 2
(c)
(e)
(g)
∫
∫
∫
∫
1
49+t 2 dt (d)
2
√
1−4t 2 dt
3
x 2 +3 dx
(f) ∫
1
2 √ 1−v dv 2
1
50+2t 2 dt
1
√
36 −9x 2 dx
7 The speed, v(t),ofaparticleis given by
v(t) =t +e −t
(a) Findthe distancetravelled by the particle.
(b) Calculate the distancetravelled betweent = 1
andt=3.
8 The capacitanceofacapacitor is 0.1F.The current,
i(t),through the capacitor is given by
i(t) = 50sinπt
Derive an expression forthe voltageacross the
capacitor.