Mathematics_Today_-_October_2016

43. [sin ]
    /

/

∫ xdx=

π

π

2

32

(where[.] is greatest integer

function)
(a) –π (b) π/2 (c) –π/2 (d) π

Matrix-Match Type

44. Match the following :

Column I Column II

A. The value of

dx

1 nx

2

+

−

∫ cot

/

α

πα

where, 0

2

<<α π,n> 0 is

P. π

2

B. The value of

sin^2 ,

1

xxdx 0

+

>

−

∫ α α

π

π

is

Q. π

α

4

−

C. The value of

sin

sin cos

2

22

3

2 n

nn

x

xx

dx

+

−

∫

α

πα

is

R. 3

4

π− 2 α

D. The value of

tan

tan cot

tan

cot x

xx

dx

− +

−

∫

1

1

α

α

is

S. π

α

4

−tan−^1

45. Match the following :

Column I Column II

A. sec

(sec tan )

x

xx

dx

+

∫ 2 =

P.

logsin

sin

x

x

− C

−

(^2) +
1
B. cos
(sin )(sin )
x
xx
dx
−−
∫ =
12
Q.
−

• cos +
  (sin)
  2
  21 2
  x
  x
  C
  C.
  sin ,
  ||
  −


• 
  

  ⎛
  ⎝⎜
  ⎞
  ⎠⎟
  <=
  ∫
  1
  2
  2
  1
  1
  x
  x
  dx
  x
  R. 2xtan–1 x – log (1 + x^2 )

  
  


• C
  D. (tanxxdx+=cot )
  ∫
  S.
  2 1
  2
  tan^1 tan
  tan
  −⎛ −
  ⎝⎜
  ⎞
  ⎠⎟+
  x
  x
  C
  Integer Answer Type

46. If the value of definite integral xa axdx

a

∫ ⋅ −[log ]

1

where a > 1, and [.]denotes the greatest integer, ise−^1

2

then the value of 5[a] is ___

47. IfIx=+∫ (sin (sin ) cos (cos ))^22 x xdx,
    0

π

then

[I] = ____, where [.] denotes the greatest integer

function

48. Area bounded by 2 ≥ max. {|x – y|, |x + y|} is
    k sq. units then k =


49. The area bounded by the curves y = ln x, y = ln|x|,
    y = |lnx|, y = |ln|x||(in sq. units) is


50. Let f(x) = x^3 + 3x + 2 and g(x) is the inverse of it.
    The area bounded by g(x), the x-axis and the ordinates
    at x = –2 and x = 6 is m
    n

where m, n ∈ N and G.C.D of

(m, n) = 1 then m – 2 =

51. The integral (| cos | sin | sin | cos )
    /

/

∫ tt t tdt+

π

π

4

54

has

the value equal to

52. If the area bounded by the curves y = –x^2 + 6x – 5,
    y = –x^2 + 4x – 3 and the line y = 3x – 15 is^73
    λ

sq. units,

then the value of λ is

53. The minimum area bounded by the function
    y = f(x) and y = αx + 9, (α ∈ R) where f satisfies the
    relation f(x + y) = f(x) + f(y) +yfx xyR() ,∀∈and
    f′(0) = 0 is 9A, then value of A is


54. Let R = {x, y : x^2 + y^2 ≤ 144 and sin(x + y) ≥ 0} and
    S be the area of region given by R, then find S/9π.


55. If the area bounded by [x] + [y] = n and y = k;
    n, k ∈ N and k ≤ (n + 1) and [.] greatest integer function,
    in the first quadrant, is n + r, then find r.
    SOLUTIONS


56. (b) : sin
    sin( )

sin( )

sin( )

x

x

dx x

x

dx

−

= − +

∫∫α −

αα

α

= − + −

∫ −

(sin( ) cos cos( )sin

sin( )

xx

x

αα ααdx

α

=+∫cosαααdx ∫sin ⋅−cot(x )dx

= xcosα + sinα · logsin(x – α) + c