THE VIDYALAY PRAKASHAN
INTEGRATION
CLASS XII
DPS - 2
Imp Formula:
2 sin A cos B = sin ( A + B ) + sin ( A − B )
2 cos A sin B = sin ( A + B ) − sin ( A − B )
2 cos A cos B = cos ( A + B ) + cos ( A − B )
2 sin A sin B = cos ( A − B ) − cos ( A + B )
Integrate each of the following trigonometric function w.r.t x :
57. sin 4 x cos 3x dx 58. cos 2x cos 4x dx
59. sin x sin 2 x sin 3x dx 60. cos x cos 2x cos 3x dx
61. 1 + sin x dx 62. 1 − sin x dx
63. 1 + cos x dx 64. 1 − cos x dx
sin 8 x − cos 8 x 1 + cos x
65. 1 − 2 sin 2 x cos 2 x dx 66. 1 − cos x dx
f ' ( x)
INTEGRALS OF THE FORM: f ( x)
dx
sec 2 x sec 2 x 2x + 6
67. tan x
dx 68. 8 + tan x dx 69. x 2 + 6 x + 7 dx
ex e−x 1
70. 1 + e x dx 71. 1 + e − x dx 72. 1 + e x dx
e x − e−x sin 2 x sin 2 x
73. x − x dx 74. sin 2 x dx 75. cos 2 x dx
e +e
sin x 1 1
76. sin (x − a ) dx 77. sin x cos x dx 78. sin 2x cos 2x dx
sin 2 x sin( x − a)
79. a 2 sin 2 x + b 2 cos 2 x dx 80. sin x
dx
1 1
81. sin( x − a) sin( x − b) dx 82.
cos( x − a) cos( x − b)
dx
1 1
83. sin( x − a) cos(x − b) dx 84.
cos( x − a) sin( x − b)
dx
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THE VIDYALAY PRAKASHAN
STANDARDS FORM:
85. tan x dx 86. cot x dx 87. sec x dx
1 1
88. cosec x dx 89. x log x dx 90. x(15 + log x) dx
x cosec x
sin x
91. 1 + x dx 92. x
dx 93.
x
dx
log tan
2
1 1 + tan x sec x
94. x ( x +1 ) dx 95. x + log sec x dx 96. log (sec x + tan x ) dx
x +1 1 − sin 2 x 1
97. x(x + log x) dx 98. x + cos 2 x dx 99. x log x log (log x) dx
1 + tan x
100. 1 − tan x dx
f ' ( x)
[ f ( x)]n dx , f f
' '
INTEGRALS OF THE FORM: ( x) f ( x) dx , ( x)[ f ( x)]n dx
sec 2 x cosec 2 x log 3 x
101. tan 9 x dx 102. cot 5 x dx 103. x dx
x
log tan
104.
2
dx 105.
2 + log x
dx 106.
2
(
sec 2 tan −1 x
dx
)
sin x x 1+ x2
tan x sec 2 x 2x
108. 2 2 2 2 2 x dx
x
107. (a + b tan x) 2 2
dx
1 + sin x
109. dx 110. sec x log (sec x + tan x ) dx
x − cos x
x2
111. cosec x log (cosec x − cot x) dx 112. dx
1+ x
113. tan 3 x dx 114. tan 4 x dx
sec 4 x
115. dx 116. sec3 x tan x dx
tan x
1
117. sec n x tan x dx 118. dx
sin x sin (x + )
3
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