 wOHhk fmdÿ iy;sl m;% (Wiia fm<) úNd.

h 

ixhqla; .Ks;h

 wkql,kh 

Manoj Solangaarachchi
(B. Sc.)

(01) yn = sinn x hehs .ksuq' fuys n ´kEu mQ¾K ixLHdjls'

d2yn = n (n – 1) yn – 2 – n2yn nj fmkajkak'
dx2


In =
0
a –x
yn dx hehs ,shuq' (n  1)

In =
0
 a– x d2yn dx nj fmkajkak'
dx2

(01) ta khska" In = n (n – 1) In – 2 nj fmkajkak' I4 ys w.h wfmdaykh lrkak'

n2 + 1
(w'fmd'i'W'fm' – 1986)

---------------------------------------------------------------------------------

a a

  (x – a)2 dx hehs .ksuq'

2
(02) (i) I = x dx iy J =
x2 + (x – a)2 x2 + (x – a)2
0 0

(02) (i) I = J = a nj fmkajkak'

2
2


–x
(ii) y = e wdfoaYfhka" e – 2x dx wkql,h w.hkak'
1 + e –x
1
(w'fmd'i'W'fm' – 1986)

---------------------------------------------------------------------------------
p

(03)
0
 sin x sin (p – x) dx = 1 (sin p – p cos p) nj fmkajkak'
2
(03) fuys p hkq ksh;hla fõ'
p p

(03) I =
0
  (x) dx yd J =
0
  (p – x) dx hehs .ksuq'

- 1 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

(03) fuys  (x) hkq x ys wkql,H Ys%;hla o p hkq Ok ksh;hla o fõ' I = J nj
fmkajkak'

f (x) hkq x ys ish¨ ;d;a;aúl w.hka i|yd f (x) + f (p – x) = q jk whqßka jQ x

ys wkql,H Ys%;hls' fuys p (  0) iy q ksh;hka fõ'
p

(i)
0
 f(x) dx = 1 pq
2
p

(ii)
0
 sin x sin (p – x) f(x) dx = 1 q (sin p – p cos p) nj fmkajkak'
4
(w'fmd'i'W'fm' – 1987)

---------------------------------------------------------------------------------

(04) (i) f(x) hkak Nskak Nd. weiqfrka m%ldY lrkak'

fuys f(x) = 2 fõ'
(x + 1)2 (x2 + 1)
k

0
 f(x) dx, (k  0) w.hkak'
k

k
lim
 0
 f(x) dx mßñ; nj wfmdaykh lrkak'

/2

ta khska fyda wka l%uhlska fyda

0
 d
(1 + tan )2
w.hkak'

 

(ii) I =


sin (log x) dx iy J =


cos (log x) dx hhs .ksuq'

fuys 0     fõ'

(ii) I + J =  sin (log ) –  sin (log ) nj fmkajkak' fuh Wmfhda.S lr

.ksñka yd I iy J ys ;j;a f¾Çh ixfhdackhla i,lñka I yd J w.hkak'

(w'fmd'i'W'fm' – 1988)

---------------------------------------------------------------------------------

(05) fldgia jYfhka wkql,kh Wmfhda.S lr.ksuka"

 sin (log x) dx w.hkak'

- 2 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 r ksh;hla jk I =
 xr sin (log x) dx iy J =
 xr cos (log x) dx kï"

1 + r I – r J = xr + 1 {sin (log x) – cos (log x)} + ksh;hla nj idOkh

2 2 2
lrkak'

xr + 1 sin (log x) wjl,kh lsÍfuka fyda wka l%uhlska I iy J w;r ;j;a

iïnkaOhla ,ndf.k I = xr + 1 {(r + 1) sin (log x) – cos (log x)} + ksh;hla
r2 + 2r + 2
nj wfmdaykh lrkak'

e
ax
ta khska" a yd b ksh; jk sin bx dx w.hkak'
(w'fmd'i'W'fm' – 1989)

---------------------------------------------------------------------------------

(06) (i) x (1 – x)2 = (1 + x2)(x – 2) + 2 iy

x4 (1 – x)4 = (1 + x2)(x6 – 4x5 + 5x4 – 4x2 + 4) – 4 nj i;Hdmkh lrkak'

by; oelafjk m%;sM,h Ndú;d lsÍfuka"

1 1

 x (1 – x) dx iy
 x4 (1 – x)4 w.hkak'
2

1 + x2 1 + x2
0 0

3    22 nj wfmdaykh lrkak'
7

(ii) n Ok ksÅ,hla jk úg"

 sin3  cosn  d = 2
3+n  sin  cosn  d – sin2  cosn + 1  nj
3+n
idOkh lrkak'
/2

ta khska"
0
 sin3  cos3  d w.hkak'

0
 x3 (1 – x2) dx wkql,kh i,ld ne,Sfuka" Tfí m%;sM,h ksjerÈ ±hs

fidhd n,kak'
(w'fmd'i'W'fm' – 1990)

- 3 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 a a

(07) (i)
0
 f(x) dx =
0
 f(a – x) dx nj fmkajkak'

 

ta khska"
0
 x sinn x dx = 
2 0
 sinn x dx nj fmkajkak'

fuys n hkq Ok ksÅ,hls'

 
;jo" n  2 úg" n  sin x dx = (n – 1)
n
 sinn – 2 x dx njo fmkajkak'
0 0
 
ta khska"  x sin4 x dx iy  x sin5 x dx w.hkak'
0 0

(ii) d loge (sec  + tan ) = sec  nj fmkajd"

d

 dy
 y2 – 1
fiùug th Ndú;d lrkak'

-1
 dx
(x + 2) 2x2 + 6x + 5
we.ehSu i|yd y = 2x2 + 6x + 5 wdfoaYh
x+2

Ndú; lrkak'
(w'fmd'i'W'fm' – 1991)

---------------------------------------------------------------------------------
c

(08) (a) a  0 kï" d (ax) ,nd f.k" c ksh;hla úg"

dx 0
 ax dx w.hkak'
ax + 1
c c

0  c   úg" I =
2 -c
 cos x dx iy J =
1 + ax -c
 ax cos x dx kï"
1 + ax

(i) t = – x wdfoaYh fh§fuka fyda wka l%uhlska fyda I = J nj

fmkajkak'

(ii) I = J ,nd .kak' tkhska" c = /6 úg J ys w.h ,shkak'

1
(06) (b)
0
 dx
(2 + x)1/2 (2 – x)3/2
w.hkak'

(w'fmd'i'W'fm' – 1992)

- 4 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

(09) (a) u = 1 – x wdfoaYfhka fyda wka l%uhlska fyda"
x

 (1 + x2) dx wkql,kh w.hkak'

1 + x4

(06) (b) n Ok ksÅ,hla hehs is;uq'

/2 /2

 
sin (2n + 1) x dx – sin (2n – 1) x dx = 0 nj fmkajd"
sin x sin x
0 0

/2

0
 sin (2n + 1) x dx =  nj wfmdaykh lrkak'
sin x 2

/2 /2

;jo"
0
 sin2 (n + 1) x dx –
sin2 x 0
 sin2 nx dx =  nj fmkajd"
sin2 x 2

/2

0
 sin2 (n + 1) x dx ys w.h wfmdaykh lrkak'
sin2 x
(w'fmd'i'W'fm' – 1994)

---------------------------------------------------------------------------------

(10) (i)
 5x + 3
(x – 1)(x + 1)2
dx fidhkak'

(ii) x + 1 = 1 wdfoaYfhka"
t
3 1/2


1
dx
(x + 1)(4x – 3 – x )
2 1/2
=
1/4
 dt
{(4t – 1)(1 – 2t)} 1/2
nj fmkajkak'

(ii) t = 1 cos2  + 1 sin2  fh§fuka fyda wka l%uhlska fyda wkql,fha w.h
4 2
(ii)  nj fmkajkak'
22

/2
(iii)
0
 sin5  cos4  d = 8 nj fmkajkak'
315

(w'fmd'i'W'fm' – 1995)

- 5 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

(11) (i) wfoaY lsÍfï l%uh Ndú;fhka fyda wkal%uhlska fyda


𝑑𝑥
fidhkak'
(𝑥 + 1)√𝑥 2 + 3𝑥 + 1

(06) (ii) fldgia jYfhka wkql,kh Ndú;fhka fyda wkal%uhlska fyda

 x3 tan-1 x dx ,nd .kak'


4
(06) (iii)  cos 2x tan3 x dx w.hkak'
0
(w'fmd'i'W'fm' – 1996)

---------------------------------------------------------------------------------

(12) (a) f iy g hkq [– a, a] m%dka;rh u; wkql,H Ys%; folla hehs is;uq'

[– a, a] ys ish¨ u x i|yd f(– x) = f(x) iy g(– x) = – g(x) hehs is;uq'
a a a

(28) (c)
-a
 f(x) dx = 2,
-a
 f(x) dx iy 
-a
g(x) dx = 0 nj fmkajkak'


𝑥2 + 𝑥3
3 dx w.hkak'
(4 − 𝑥 2 )2
-1

(28) (b) fldgia jYfhka wkql,kh fh§fuka"

a

 x2 h (x) dx = a2 h (a) – 2ah (a) + 2h (a) – 2h(0) nj fmkajkak;

0
𝑑ℎ 𝑑2 ℎ 𝑑3ℎ
fuys h = , h (x) = iy h (x) = .
𝑑𝑥 𝑑𝑥 2 𝑑𝑥 3


𝑥2
5 dx w.hkak'
(𝑥 − 1)2
0
(w'fmd'i'W'fm' – 1997)

---------------------------------------------------------------------------------

1
(13) (a) (𝑥 2 − 1)(𝑥 2 − 3𝑥 + 2)
Nskak Nd. f,i m%ldY lrkak'

(13) (a) ta khska  𝑑𝑥

(𝑥 2 − 1)(𝑥 2 − 3𝑥 + 2)
fidhkak'

a a
(28) (b)  f(x) dx =  f(a – x) dx nj fmkajd"
0 0

- 6 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

  
2 2

 
𝑥 𝑑𝑥 𝜋 𝑑𝑥
ta khska" = nj fmkajkak'
sin 𝑥 + cos 𝑥 4 sin 𝑥 + cos 𝑥
0 0

2


𝑥 𝑑𝑥 𝜋
= ln (√2 + 1) nj wfmdaykh lrkak'
sin 𝑥 + cos 𝑥 2√2
0

(w'fmd'i'W'fm' – 1998)

---------------------------------------------------------------------------------

(14)  1
2 + sin 𝑥
dx fidhkak' bÕsh t = tan
𝑥
2
fhdod n,kak'

cos2 𝑥 𝐶
= A + B sin x + jk mßÈ A, B, C ksh; ks¾Khlr"
2 + sin 𝑥 1 + sin 𝑥


2


cos2 𝑥
ta khska" dx w.hkak'
2 + sin 𝑥
0

2


1
sin x ln (2 + sin x) dx = ln 2 +  [ 1 − ] – 1 nj wfmdaykh lrkak'
√3
0

(w'fmd'i'W'fm' – 1999)

---------------------------------------------------------------------------------


1
(15) (a) iqÿiq wdfoaYhla Wmfhda.S lr.ksñka" 4 2 dx w.hkak'
(𝑥 3 + 𝑥 3 )
1

 
(15) (b) I=  e –2x
cos x dx yd J =  e–2x sin x dx hehs .ksuq'
0 0

(15) (b) fldgia jYfhka wkql,k l%uh Wmfhda.S lr.ksñka" I = 2J yd

J = 1 + e–2 – 2I nj fmkajkak'

(15) (b) ta khska" I yd J ys w.hka fidhkak'

(15) (c)  𝑥 2 − 5𝑥
(𝑥 − 1)(𝑥 + 1)2
dx fidhkak'

(w'fmd'i'W'fm' – 2000)

---------------------------------------------------------------------------------

- 7 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 √2

(16) (a) iqÿiq wdfoaYhla fh§fuka" 

1
1
𝑥 2 √4 − 𝑥 2
dx wkql,h w.hkak'

(28) (b) fldgia jYfhka wkql,k l%u Ndú;fhka"

4

 x ln x dx = a ln b + c nj fmkajkak'
2
fuys a, b iy c hkq ks¾Kh l< hq;= ksÅ, fõ'
1

28) (c)  (7𝑥 − 𝑥 2 )

(2 − 𝑥)(𝑥 2 + 1)
dx fidhkak'
0 (w'fmd'i'W'fm' – 2001)

---------------------------------------------------------------------------------

(17) (a) iqÿiq wdfoaYhla fh§fuka" 

1
𝑥3
√𝑥 2 − 1
dx wkql,h w.hkak'

(28) (b) fldgia jYfhka wkql,k l%uh Ndú;fhka"

1

 x tan-1 x dx wkql,h w.hkak'

0


5𝑥 − 4
28) (c) (1 – 𝑥 + 𝑥 2 )(2 + 𝑥)
dx fidhkak'
1 (w'fmd'i'W'fm' – 2002)

---------------------------------------------------------------------------------


𝑑𝑥
(18) (a) iqÿiq wdfoaYhla fh§fuka" 3 wkql,h w.hkak'
1 + √𝑥
1

(28) (b) fldgia jYfhka wkql,kh Ndú;fhka"

1

 x2 e2x + 3 dx wkql,h w.hkak'

0


𝑑𝑥
28) (c) dx fidhkak'
𝑥 (𝑥 2 + 3)
(w'fmd'i'W'fm' – 2003)

- 8 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

---------------------------------------------------------------------------------

23


𝑑𝑥
(19) (a) iqÿiq wdfoaYhla fhdod .ksñka" w.hkak'
(𝑥 + 1)√2𝑥 + 3
11

(28) (b) fldgia jYfhka wkql,kh Ndú;fhka"  e3x cos 4x dx fidhkak'

28) (c)  sin4 2x dx fidhkak'

(w'fmd'i'W'fm' – 2004)

---------------------------------------------------------------------------------

2


𝑥 𝑑𝑥
(20) (a) tan = t wdfoaYh fhdod .ksñka" wkql,h w.hkak'
2 5 + 4 sin 𝑥
0

(28) (b) fldgia jYfhka wkql,kh fhdod .ksñka"  15x3 √1 + 𝑥 2 dx wkql,h

w.hkak'

28) (c)  𝑥 2 − 10𝑥 + 13

(𝑥 − 2)(𝑥 2 − 5𝑥 + 6)
dx fidhkak'
(w'fmd'i'W'fm' – 2005)

---------------------------------------------------------------------------------

2


𝑑𝑥
(21) (a) iqÿiq wdfoaYhla fhdod .ksñka" w.hkak'
3 + 2 cos 𝑥 + sin 𝑥
0

(28) (b) fldgia jYfhka wkql,kh Ndú;fhka"  e4x sin 3x dx fidhkak'


𝑑𝑥
28) (c) Nskak Nd. Ndú;fhka" fidhkak'
𝑥3 + 1
(w'fmd'i'W'fm' – 2006)

---------------------------------------------------------------------------------


𝑥3 + 1
(22) (a) Nskak Nd. Wmfhda.S lr.ñka" dx fidhkak'
𝑥 (𝑥 − 1)3

(28) (b) 25 cos x + 15  A (3 cos x + 4 sin x + 5) + B (– 3 sin x + 4 cos x) + C jk

wdldrhg A, B yd C fidhkak'


- 9 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -
 25 cos 𝑥 + 15
ta khska" dx fidhkak'
3 cos 𝑥 + 4 sin 𝑥 + 5

28) (c) fldgia jYfhka wkql,kh Wmfhda.S lr.ksñka"

  
2 2 2

 sin x dx =  
6 5 5.3 5𝜋
sin4 x dx = sin2 x dx = nj fmkajkak'
6 6.4 32
0 0 0

6

ta khska"  sin6 3x dx w.hkak'

0
(w'fmd'i'W'fm' – 2007)

---------------------------------------------------------------------------------


𝑑𝑥
(23) (a) Nskak Nd. Wmfhda.S lr.ñka" fidhkak' fuys a  0 fõ'
(𝑥 2 − 𝑎2 )2

𝑑 2𝑥
(28) (b) (i) [ ] = 2x nj fmkajkak'
𝑑𝑥 ln 2

(ii)  2x dx nj fmkajkak'
1

(iii) fldgia jYfhka wkql,kh Ndú;fhka"  2√𝑥 + 1 dx w.hkak'

-1

(w'fmd'i'W'fm' – 2008)

---------------------------------------------------------------------------------


𝑒𝑡
(24) (a) Ik = dt hehs .ksuq' fuys t  0 jk w;r k Ok mQ¾K ixLHdjls'
𝑡𝑘

𝑒𝑡
(k – 1) Ik – Ik – 1 + = C nj fmkajkak' fuys C wNsu; ksh;hls'
𝑡𝑘 − 1


1−𝑥 2
ex [ ] dx fidhkak' fuys x  – 1 fõ'
1+𝑥

(28) (b) f hkq ;d;a;aúl ixLHd l=,lh u; w¾: olajd we;s ;d;a;aúl w.hka
a

.kakd Ys%;hla jk w;r" J =

0
 f(x) dx fõ' fuys a  0 fõ'
a


0
f(a – x) dx = J nj fmkajkak'


2


- 10 - |Combined Maths | Manoj Solangaarachchi - B.Sc.
0
– wkql,kh -
 sin2𝑘 𝑥
dx w.hkak' fuys k Ok mQ¾K ixLHdjls'
cos2𝑘 𝑥 + sin2𝑘 𝑥

(w'fmd'i'W'fm' – 2009)

---------------------------------------------------------------------------------


2𝑥
(25) (a) Nskak Nd. Wmfhda.S lr.ñka" (1 + 𝑥 2 )(1 + 𝑥)2
dx fidhkak'

(28) (b) I =  eax cos bx dx yd J =  eax sin bx dx hehs .ksuq; fuys a yd b hkq

Y=kH fkdjk ;d;a;aúl ixLHd fõ'

(i) bI + aJ = eax sin bx,
(ii) aI – bJ = eax cos bx nj fmkajkak'
ta khska" I yd J fidhkak'

28) (c) x3t + 1 = 0 wdfoaYh Wmfhda.S lr.ksñka fyda fjk;a wdldrhlska fyda"
1
−
2


𝑑𝑥 1 9
= ln [ ] nj fmkajkak'
𝑥 (𝑥 3 − 1) 3 2
-1
(w'fmd'i'W'fm' – 2010)

---------------------------------------------------------------------------------
e
3
(26) (a) fldgia jYfhka wkql,kh fh§fuka"  𝑥 2 ln x dx w.hkak'
1

(28) (b) t = tan x hehs .ksuq'

1 − 𝑡2 2𝑡 𝑑𝑥 𝑡
cos 2x = , sin 2x = yd = nj fmkajkak'
1 + 𝑡2 1 + 𝑡2 𝑑𝑡 1 + 𝑡2


2


1 1
ta khska" dx = nj fmkajkak'
4 cos 2𝑥 + 3 sin 2𝑥 + 5 12
0

28) (c) a yd b hkq m%Nskak ;d;a;aúl ixLHd hehs .ksuq'

1 𝐴 𝐵
28) (c) x   – {a, b} i|yd (𝑥 − 𝑎)(𝑥 − 𝑏)
= + jk whqßka A yd B ksh;
𝑥−𝑎 𝑥−𝑏
fidhkak'
1
by; iólrKfha x, a yd b iqÿiq f,i m%;sia:dmkh lrñka" (𝑥 2 + 𝑎2 )(𝑥 2 + 𝑏2 )

- 11 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 hkak Nskak Nd. weiqfrka ,shd olajd"


1
ta khska" (𝑥 2 + 𝑎2 )(𝑥 2 + 𝑏2 )
dx fidhkak'

(w'fmd'i'W'fm' – 2011)

---------------------------------------------------------------------------------

4
(27) (a)  (sin3 x – cos3 x) dx =
3
nj fmkajkak' (fuu W;a;rh fodaI iys;h')
0

(28) (b) fldgia jYfhka wkql,kh fhdod .ksñka fyda fjk;a wdldrhlska fyda

 x3 tan-1 x dx fidhkak'


2𝑥 2 − 3
28) (c) Nskak Nd. fhdod.ksñka" dx fidhkak'
(𝑥 − 2)2 (𝑥 2 + 1)

(w'fmd'i'W'fm' – 2012)

---------------------------------------------------------------------------------

(28) (a) fldgia jYfhka wkql,kh Ndú;fhka"  x2 sin–1 x dx fidhkak'

(28) (b) Nskak Nd. Ndú;fhka"

 x2 + 3x + 4 dx fidhkak'
(x2 – 1)(x + 1)2

(28) (c) a2 + b2  1 jk mßÈ a, b   hehs o"

/2

(ii) I =
0
 2
a + cos x
2
a + b + a cos x + b sin x
dx yd

/2

(ii) J =
0
 b + sin x
a2 + b2 + a cos x + b sin x
dx hehs o .ksuq.

(ii) aI + bJ = π nj fmkajkak.
2

(ii) bI – aJ ie,lSfuka I yd J ys w.hka fidhkak.

(w'fmd'i'W'fm' – 2013)

---------------------------------------------------------------------------------

- 12 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

(29) (a)  3𝑥 + 2
𝑥 2 + 2𝑥 + 5
𝑑𝑥 fidhkak'

e


1
(28) (b) fldgia jYfhka wkql,kh Ndú;fhka" cos (In x) dx = − (𝑒  + 1) nj
2
1
fmkajkak'

a a
(28) (c)  f(x) dx =  f(a – x) dx iQ;%h msysgqjkak; fuys a hkq ksh;hls'
0 0

2

(28) p(x) = (x – )(2x + ) hehs o I =

0
 𝑠𝑖𝑛2 𝑥
𝑝(𝑥)
𝑑𝑥 hehs o .ksuq'


2


𝑐𝑜𝑠 2 𝑥
by; m%;sM,h Ndú;fhka I = 𝑑𝑥 nj fmkajkak'
𝑝(𝑥)
0

2


1 1
I i|yd jQ by; wkql, fol Ndú;fhka I = 𝑑𝑥 nj wfmdaykh
2 𝑝(𝑥)
0
lrkak'

1 1
ta khska" I = In ( ) nj fmkajkak'
6 4
(w'fmd'i'W'fm' – 2014)

---------------------------------------------------------------------------------
 
(30) (a)  f(x) dx =  f( – x) dx nj fmkajkak'
0 0

2



sin2 x dx = nj;a fmkajkak'
4
0



2
ta khska" x sin2 x dx = nj fmkajkak'
4
0


2
(28) (b) iqÿiq wdfoaYhla yd fldgia jYfhka wkql,k l%uh Ndú;fhka" 𝑥 3 𝑒 𝑥 𝑑𝑥
fidhkak'

1 𝐴 𝐵𝑥 + 𝐶
(28) (c) = + jk mßÈ A, B yd C ksh;j, w.hka fidhkak'
𝑥3 − 1 𝑥−1 𝑥2 + 𝑥 + 1

- 13 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 1
ta khska" hkak x úIhfhka wkql,kh lrkak'
𝑥3 − 1

2


𝑥 𝑑𝑥 1
(28) (d) t = tan wdfoaYh Ndú;fhka" = nj fmkajkak'
2 5 + 4 cos 𝑥 + 3 sin 𝑥 6
0

(w'fmd'i'W'fm' – 2015)

---------------------------------------------------------------------------------

(31) (a) (i)  𝑑𝑥

√3 + 2𝑥 − 𝑥 2
fidhkak'

𝑑 𝑥−1
(ii) (√3 + 2𝑥 − 𝑥 2 ) fidhd" ta khska" 𝑑𝑥 fidhkak'
𝑑𝑥 √3 + 2𝑥 − 𝑥 2


𝑥+1
by; wkql, Ndú;fhka √3 + 2𝑥 − 𝑥 2
𝑑𝑥 fidhkak'


2𝑥 − 1 (2𝑥 – 1)
(28) (b) (𝑥 + 1)(𝑥 2 + 1)
Nskak Nd. weiqfrka m%ldY lr" ta khska" (𝑥 + 1)(𝑥 2 + 1)
𝑑𝑥
fidhkak'

(31) (c) (i) n  – 1 hehs .ksuq' fldgia jYfhka wkql,kh Ndú;fhka"  𝑥 𝑛 (In x) dx
fidhkak'
3

(ii)
1
 𝐼𝑛 𝑥
𝑥
𝑑𝑥 w.hkak'
(w'fmd'i'W'fm' – 2016)

---------------------------------------------------------------------------------

(32) (a) (i)

𝑥 (𝑥 + 1)2
1
Nskak Nd. weiqfrka m%ldYlr" ta khska"
 1
𝑥 (𝑥 + 1)2
dx fidhkak'

(ii) fldgia jYfhka wkql,kh Ndú;fhka"  xe –x dx fidhd"

ta khska" y = xe –x jl%fhka o x = 1" x = 2 yd y = 0 ir, f¾Ldj,ska o

wdjD; fmfofiys j¾.M,h fidhkak'
c

 ln (𝑐 + 𝑥)
(28) (b) c  0 yd I = dx hehs .ksuq' x = c tan  wdfoaYh Ndú;fhka"
𝑐 2 + 𝑥2
0

4


𝜋 1
I= ln c + J nj fmkajkak; fuys J = ln (1 + tan ) d fõ'
4𝑐 𝑐
0

- 14 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 a a

 
𝜋
a ksh;hla jk f(x) dx = f(a – x) dx iQ;%h Ndú;fhka" J = ln 2 nj
8
0 0

fmkajkak'
𝜋
I= ln (2c2) nj wfmdaykh lrkak'
8𝑐

(w'fmd'i'W'fm' – 2017)

---------------------------------------------------------------------------------

(33) (a) (i) x2, x1 yd x0 ys ix.=Kl iei£fuka"

ish¨ x   i|yd Ax2 (x – 1) + Bx (x – 1) + C (x – 1) – Ax3 = 1 jk mßÈ"
A, B yd C ksh;j, w.hka fidhkak'


1 1
ta khska" hkak Nskak Nd. j,ska ,shd olajd dx
𝑥 3 (𝑥 − 1) 𝑥 3 (𝑥 − 1)

fidhkak'

(ii) fldgia jYfhka wkql,kh Ndú;fhka"  x2 cos 2x dx fidhkak'


sin 𝑥
(28) (b)  = tan-1 (cos x) wdfoaYh Ndú;fhka" √1 + cos2 𝑥
dx = 2 ln (1 + √2) nj
0
fmkajkak'
a a
a ksh;hla jk  f(x) dx =  f(a – x) dx iQ;%h Ndú;fhka"
0 0



0
𝑥 sin 𝑥
√1 + cos2 𝑥
dx fidhkak'

(w'fmd'i'W'fm' – 2018)

---------------------------------------------------------------------------------

- 15 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 4

√
𝜋 𝑥−3
(34) (a) 0 i|yd x = 2 sin2  + 3 wdfoaYh Ndú;fhka dx w.hkak'
4 5−𝑥
3


1
(28) (b) Nskak Nd. Ndú;fhka" (𝑥 − 1)(𝑥 − 2)
dx fidhkak'


1
t  2 i|yd f(t) = (𝑥 − 1)(𝑥 − 2)
dx hehs .ksuq'
3

t  2 i|yd f(t) = In (t – 2) – In (t – 1) + In 2 nj wfmdaykh lrkak'

(28) (b) fldgia jYfhka wkql,kh Ndú;fhka"  In (x – k) dx fidhkak'

(28) (b) fuys k hkq ;d;a;aúl ksh;hls'

(28) (b) ta khska"  f(t) dt fidhkak'

b b

(28) (c) c yd b ksh; jk

a
 f(x) dx = 
a
f(a + b – x) dx iQ;%h Ndú;fhka"

 

 
cos2 𝑥 𝑒 𝑥 cos2 𝑥
(28) (c) 1 + 𝑒𝑥
dx =
1 + 𝑒𝑥
dx nj fmkajkak'
– –


cos2 𝑥
(28) (b) ta khska" dx ys w.h fidhkak'
1 + 𝑒𝑥
–

(w'fmd'i'W'fm' – 2019)



Manoj Solangaarachchi

- 16 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -

 (B. Sc.)

- 17 - |Combined Maths | Manoj Solangaarachchi - B.Sc. – wkql,kh -