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Class- XII
Subject - Mathematics
INDEFINITE INTEGRAL
1MARK
1.
e
2 logcos x
dx
2.
sin(tan 1 x)
1  2 tan x(tan x  sec x) dx
dx
3.  1  x 2



x2
sin 1 x
dx
 1  x3 5. 
1 x2
4.
2
dx
.
6.

x 2  4 x  2dx
2x  1

7. ( x  x
1  sin 3 x
dx

11. 3 x  cos 3 x
12.

2
2x
1 4x
)
dx
dx
8.

9.
23.
x e
2
x
3
cos x dx
20.
dx 24. Prove that
Co sec x.Sec x.dx
17. 
2

0
Sec 2(log x)
dx
x
2
 x
.∫{f’’(ax+b)/f(ax+b)} dx 21.
a
x 10.

1
sin 2 x  2 cos x
dx
e 3 log x ( x 4 )dx
2

x  2 sin x 16.
 sin
19.
 x
x
 e sec x  log(sec x  tan x)dx 14.  sin 2 x cos 2 x dx
13.

15. 5  cos
1
dx
4 x  3x 2  5dx
a
18.

2
 a  e .a  sin a dx 22.
x
x
x
x
 x dx
1
a
f ( x)dx   f (a  x)dx
o
4MARKS
1. Evaluate :
x  sin x
dx

4. 1  cos x
dx
2.
5. 
dx
8. cos( x  a) cos( x  b) 9.
dx
sin 1 x  cos 1 x
dx .
sin 1 x  cos 1 x
3.
6. 

1 
 log log x   log x 2  dx
10.
2  sin x 2
x2  4
e dx
dx

4
2

12. x  x  16
13. 1  cos x
14.
x
 x 6(log x)
dx
 7 log x  2
2
3x  2
dx . 7.  Cos 4 xdx
( x  3)( x  1)

cot x dx
11.
ex
 6  5e x  e 2 x dx 15.


x2  1
dx
x4  1
dx
7  6 x  x 2 16.
e
dx
1
x
17.
dx

21. 

6  x  x2
e x 1
22. 
e2 x  1
 2 x dx
26. e  1 27.
18.
5x  3
x 2  4x  10
31.
dx
32.
 1  sin x 33.  x
1/ 2
x4  x2 1
dx
 2
44. x  x  1
45.
x3  x2
x4  9
cos x  sin
53. 
58. 
63.
sin 2 x

2
dx
(x 2  1)e x
dx
( x  1) 2
20.
5
55
x
x
.55 .5 x dx
( x  3)e x
 ( x 1)3 dx
25.
24.
x3

5  4 x  x 2 26.
x2  1
tan x
dx
dx
sec 3 x dx
4
2


30. sin x.cos x
x  7 x  1 29.


dx


x2 x4 1
3
4

35.
cos 2 x  cos 2
dx
cos x  cos 
3
2 x3
3
1  3 x  x 
tan
  1  3x2  dx 39.  x e cos x dx
xe2 x
dx
(1  2 x) 2 42.
1
tan x
dx
[log(log x) 
]dx
2

sin 2 x
(log
x
)
43.

dx
x2
1
dx
dx
4

2
2

( x  a)( x  b) 46. ( x  1)( x  4)
47. x  1 48.

(1  x)e x
dx
(2  x) 2
sin x cos x
1
dx 52 . 
dx
2
2
2
sin x  b cos x
a2  x2
cos x  sin x
e x 1  x e 1
tan x
 2  sin 2 x 
dx
55.
56.
dx 57. 
dx
dx
dx 54.  e x 


x
e

sin x. cos x
e x
1  sin 2 x
 1  cos 2 x 
x
sin( x  a)
1
sin 2 x
x
dx 60.  4
dx 61. 
dx 62. 
dx 59. 
dx
2
5
sin 5 x. sin 3 x
sin( x  a)
x  x 1
x( x  1)
50. 
dx
sin( x  a) cos( x  b)
51
1
1
 4 cos x  3 sin x dx 64.  (2 sin x  3 cos x)
67. 
2
a
dx 65. 
2
sin x
cos( x  a )
dx
dx 66. 
cos x
sin( x  a)
sin( x  a)
dx 68.  cos 7 x dx 69.  cos ec 6 x dx
sin( x  b)
6MARKS
sin x
dx.
1. 
sin 4 x
sin x  cos x
dx 
2
x  sin 4 x
3.

2. cos
x
dx
( x  1) ( x  2)
2
9.

cot x 

tan x dx
sin
4. 
x2
x2  1
2
dx
log x  1  2 log x dx 7. 
x4
x sin x  cos x 2
x
3x  2
3x  2
 x  1x  22 dx 10.  x  1 x 2  1 dx 11.  x 3  x 2  x  1 dx
1
dx 6. 
5.  4
x  5 x 2  16
8. 

 x 1 x  x
1
dx
 x1/ 3 34.
sin xdx
 sin( x  5) 38.
1
1

dx
log x log x 2
41.
49.  cot3 dx
23. 
dx
(3 sin   2) cos 
d
2
  4 sin 
28.
cos 4 x  cos 2 x
dx

36. sin 4 x  sin 2 x 37.
40. 
19.
 5  cos
dx
 3  5 sin x
x2  1
dx
x4  x2  1
 




1
x
dx
ax
12. 
tan   tan 3 
1
dx
d 13 
3
sin x  sin 2 x
1  tan 
DEFINITE INTEGRALS
4/6MARKS

2
x sin x cos x
dx
4
4
sin
x

cos
x
0
1. 
3
5.

n
0
2.
xTanx
dx 3.
Secx Co sec x
a
2

x cos x dx 6.  x 2  2 x  3 dx 7.
a
0
0

9.
2
1
1
1
cot x
0
0
 /2
 sin 2 x.log tan x dx
.
0
2 3 x
0
x  ax
dx
17.

1
4
19.Evaluate

2
1
x  4 x
dx
18.
cos x
 1  sin x 2  sin x dx
x
xSinx
2
0 1  Cos x

26. Evaluate
3
x
2
x Sin x dx
24.
dx
2


1
1  cos x

0
1
2
xdx
cos x  b 2 sin 2 x
2

 (2 x  5) dx 23.Evaluate:
1
 x
0
2
20.  log (1  cos x) dx .21. 
1
3
a
0
2
  x  1  x  x  1 dx
28.
 1  sin xdx
0
22.Evaluate as limit sum

2

3
27.Evaluate :-
2
dx
0 1  tan x 15.
14.

4 x
2
 ( x  x)dx as the limit of sums.
25. Evaluate
log( 1  tan x)dx
0
dx as a limit of a sum11.
cos x
0 1  sin 2 xdx
13.
3
x

0

3

4
0
  x  3  x  4  x  5 dx
a
ax
dx  a 8.
ax

dx 10. Evaluate  e
6
16.

sin x  cos x
dx 4.
9  16sin 2 x

2

12.
 /4
 2x
3
1

2

 3x  5 dx as limit of a sum.

 5 x  1 dx as a limit of a sum.
1
APPLICATIONS OF DEFINITE INTEGRALS
1. Using integration find the area of the region bounded by the triangle whose vertices are (-1,1), (0,5)
and (3,2).
x y
x2 y2
2. Find the area of smaller region bounded by the ellipse

 1 and the straight line   1
4 3
16 9
Sketch the region enclosed between circles x2 + y2 = 1 and x2 + (y – 1)2 = 1.
Also find the area of region using integration
4. Find the area of the region enclosed between the two curves x 2  y 2  1 and ( x  1) 2  y 2  1
5. Find the area of the region  x, y  : x 2  y 2  1  x  y
6. Sketch the region common to the circle x 2  y 2  16 and the parabola x 2  6 y. Also, find the area of
the region, using integration.
7. Make the rough sketch and find the area of the region (using integration) enclosed by the curves
y2=4ax and x2=4ay.
8. Make the rough Sketch and find the area of the region
x, y  : x 2  y 2  2ax, y 2  ax, x  0, y  0
9. Draw a rough Sketch of the region x, y  : y 2  4 x, 4 x 2  4 y 2  9
DIFFERENTIAL EQUATIONS
4/6 MARKS
3.


1) Solve the differential equation xdy  ydx  x 2  y 2 dx
dy
2) Solve the differential eqn, given that y=1 when x=2: x  y  x 3
dx
3) Find the differential equation of the family of the curve given by x 2  y 2  2ax
4) Solve tany dx + sec2y tanx dy =0.
5)
Solve: (x2-y2) dx
+ 2xy dy = 0.
6) Solve : x dy/dx + y = y2 log x
7) Show that y=cos (cosx) is a solution of the differential equation:d2y
dy
 Cotx  ySin 2 x  0
2
dx
dx
8) Solve the differential equation y e y dx  y 3  2 xe y dy
dx
9) Solve the differential equations x
+ y = x cos x + sin x given that y(2) = 1
dy


10) Solve the differential equation ydx  ( x  2 y 2 )dy  0
dy
 
 y cot 2 x
given y   2
11) Solve the differential
dx
4
dy
12)Solve that differential equation x 2  1  2x  2 y  2x  1
dx
dy
 4 y  e 2 x
13)Solve the differential equation:
dx
14)Solve differential equation y dx – x dx = xy dx
dy 2
1
15)find the solution of
+ y  2 ,7  x  0 which satisfies y(2) = 2;
dx x
x
16)Solve differential equation Sec2y . (1+x2) dy + 2x tan y dx = 0given that y 

4
when x = 1
17)If y= (sin-1x)2, Prove that (1  x 2 )
d 2 y xdy

20
dx 2
dx
18.)Solve the differential equation x 2
dy 2
 x  2 y 2  xy
dx
19) Solve dy/dx = xy + y + x + 1.
20) dy/dx = y + xex
 dy 
21)Solve Sin 1    x  y
 dx 
22)Find the differential equation of the system of parabola’s having latus-rectum 7 and axis parallel to xaxis.
23)Solve the differential equation x
dy
 
 y  xCosx  Sinx given that y   =1.
dx
2
24. Solve (x2 +xy ) dy = (x2+y2).
25. cos x(1 + cos y)dx – sin y(1 + sin x)dy = 0
26. y - x dy/dx = a(y2 + dy/dx)
27. dy/dx = y cot2x , given y(/4) = 2
28. (3xy + y2 )dx + (x2 + xy)dy = 0
29. (x3 + y3 )dy = x2 y dx
30.{xcos(y/x)}(ydx + xdy) = {ysin(y/x)}(xdy – ydx)
31. dy/dx – 2y = cos3x
32. (1 + y2 )dx = (tan-1 y – x)dy
33.dy/dx + ycotx = 2x + x2 cotx , guven that y(0) = 0
34. (1 + x²)dy/dx + 2xy = 4x² given y(0) = 0
35.(1 + x²)dy/dx + 2xy = (x² + 4)1/2
36.secx dy/dx = y + sinx
37.(x(x² + y²)1/2 - y² )dx + xy dy = 0
38.Find the equation of the curve whose differential equation is (1 + y²)dx – xy dy = 0 and passes
through the point (1,0).
39.tan y dy/dx = sin(x + y) + sin(x – y)
40. Solve e2x – 3y dy + e2y – 3x dx = 0
.