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

462 Chapter 14 Techniques of integration
EXERCISES14.2
1 Useintegration by partsto findthe following:
∫
∫
(a) xsin(2x)dx (b) te 3t dt
(c)
(e)
∫
xcosxdx
∫ x
e x dx
(d)
∫
( )
v
2vsin dv
2
2 Useintegration by partsto find
∫
(a) tlntdt
∫
(b) lntdt
∫
(c) t n lntdt (n ≠ −1)
∫
(d) tsin(at +b)dt a,bconstants
∫
(e) te at+b dt a,bconstants
3 Evaluate the followingdefiniteintegrals:
∫ 1
∫ π/2
(a) xcos2xdx (b) xsin2xdx
0
0
∫ 1
∫ 3
(c) te 2t dt (d) t 2 lntdt
−1
1
∫ 2 2x
(e) dx
0 e2x 4 Find
∫
(a) t 2 e 2t dt
(b)
(c)
∫
t 2 cos3tdt
∫ ( )
t 2 t
sin dt
2
5 Evaluate the following definiteintegrals:
∫ 2
(a) t 2 e t dt
0
∫ 1
(b) t 2 sintdt
−1
∫ 1
(c) t 2 cos3tdt
0
6 Obtainareduction formulafor
∫
I n = t n e kt dt n,k constants
∫
Hencefind
∫
t 2 e 3t dt,
∫
t 3 e 3t dt and
t 4 e 3t dt.
7 Useintegration byparts twiceto obtain areduction
formulafor
∫ π/2
I n = t n sintdt
0
∫ π/2 ∫ π/2
Hencefind t 3 sintdt, t 5 sintdt
0 0
∫ π/2
and t 7 sintdt.
0
8 Useintegration byparts to find
∫ π/2
e 2x cosxdx
0
Solutions
sin2x
1 (a) − xcos2x
4 2
( )
(b) e 3t t
3 − 1 +c
9
+c
(c) cosx+xsinx+c
( ) ( )
v v
(d) 8sin −4vcos +c
2 2
(e) −e −x (x +1) +c
2 (a)
t 2 lnt
2
− t2 4 +c
(b) tlnt−t+c
(c)
(d)
(lnt)t n+1
−
tn+1
n +1 (n +1) 2 +c
sin(at +b)
a 2
−
(e) e at+b (
t
a − 1 a 2 )
+c
tcos(at +b)
a
+c