Indefinite Integration (After First Lecture) English

Mathematics

Target IIT JEE 2013 XII XII

VERY ELEMENTARY INTEGRATION

Find the antiderivative/primitive/integrals antiderivative/primitive/int egrals of the following by simple manipulation/simplifying and converting them into loving integrand i ntegrands. s. Q.1

!

x

x

2  . e  dx  dx e 5 !n x # e 4 ! n x

Q.5

!

Q.9

! 4 cos 2 · cos x · sin

Q.11

Q.14

 dx  dx

e 3 !n x # e 2 !n x

x

! !

Q.2

 dx ! 1 " cos 2x  dx

Q.6

!

21 2

Q.24

x 2x " 1

 dx  dx

Q.15

sin x " cos x

Q.26 Q.28 Q.29

Q.31

1 " sin 2x

! x 2 " 1 dx

!

sec 2x # 1 sec 2x " 1

2

!

Q.21

sin 6 x " cos 6 x 2

sin x . co cos x

 dx  dx

Q.10

!

Q.12

!

3

3

sin 2 x cos2 x

dx

dx

x#2

!

Q.25

2

$1 " x % 2

(1 " x)

!x

Q.13

 dx  dx

Q.17

cos 2x # cos 2& cos x # cos &

x " x "1 4

! x2

Q.8

cos xº dx

!

Q.22

dx

1 " 2x

dx

(2 + 2 sin 2x) dx

cos x " sin x

!

Q.19

sin x " cos x

cos x # sin x

!

Q.16

cos 2 x sin 2 x

2x # 1

! 1 " cot 2 x  dx

Q.4

cos 2 x

!

)dx(a > 0)Q.7

1 " tan 2 x

2

$

1 " x2

!

e

2x

e

%

dx

#1 x

 dx

dx

2

$

2 1 " x2

$

%

 dx  dx

!

Q.23

%$

x " 1 x2 #

x

x x " x"

%

1 # sin 2x dx

dx

x

. 2 ( 9 ' x + x+ 1 cos 4x " 1 2 ( 7 ' s i n s i n " # "  dx Q.27 ! dx * * - 3  dx 0 ! / )  8 4 , )  8 4 , 2 cot x # tan x A functio function n g define defined d for all all positiv positivee real real number numbers, s, satisf satisfies ies g'(x g'(x2) = x3 for all x>0 and g (1) = 1. Compute g (4).

!

. + 1  dx 2 ( x s i n s i n ( x ) s i n & # & " # & * - 3  dx 0 )   , 2 / 2

!

. cot 2 2x # 1 1  dx c o s 8 x c o t 4 x # 0 3  dx 2 cot 2 x / 2

Q.33

!

Q.35

!

Q.38

 + e

dx (cosx + sinx > 0)

x #1

!

(e

 dx ! 1 " tan 2 x  dx

x ln a

x dx

6

Q.20

a ln x

1 # tan 2 x

Q.3

(3 sin x cos2 x # sin3 x) dx

!

Q.18

1 " cos 2 x

!

2 x 3 " 3x 2 " 4 x " 5 2x " 1

9 # 16 x

2

dx 1 " sin x

sin 2 x # sin 2k 

Q.36

Q.39

! sin x # sin k " cos x # cos k dx

Q.44

!x

ln (ex )dx

!

Q.34

Q.41

x

!

Q.32

 dx  dx

dx

!

Q.30

!

!

dx 25 " 4 x

cos x " 1 # 2 sin 2x 2

cos 4 x # sin 4 x 1 " cos 4x

$x

2

1" x

2

1 " 2 cos 5 x

dx

x2 " 3

! x (x 6

2

" 1)

Q.40

dx   Q.43

!

dx

dx (cos2x>0)

" sin 2 x% sec2 x

Q.37

cos 8 x # cos 7 x

Q.42

sin 2x " sin 5x # sin 3x

2

dx

#1 ( 1 " cos x

tan *

)  sin x

2 " 3x 2

!

$

x2 1 " x2

!

%

- dx

dx

sinx sin x cos x cos 2x cos 4x dx

ANSWER SHEET Q.1

Q.5

2x . e x 1 " !n 2 x

1

#

Q.10

sin 2x + C

Q.14

1 .x # !n (2x " 1) 1 0 3  + C 2 2 2 /

Q.17

ex + e#x + C

Q.23 Q.25 Q.28 Q.31 Q.33 Q.36 Q.39

5

#

5

3

x

"

ax !n a

Q.11

#  

 

cos3x 3

180

'

Q.15

tan x # x + C

Q.18

x+C

 + x # 2 tan x + C   Q.21

2

2

# x + C

 

Q.26

67

Q.29

5 cos8x

# x

8 3

3

1 10

"

2

x

2

 tan

sin 3x

#1

#

 

 + C

"

3x 2

2x 5

"

7 4

 

 + C

sin 2 x

" C 

  Q.34 Q.37 Q.40

Q.41

3 2  (sin x –  cos x) + (sin k + cos k)x + C

Q.42

C  – 

2 x

 +

2 1 3 x4

3 1

 – 

5 x5

Q.13

ln x + 2 tan #1 x

 

Q.16

2x + 3 ln (x # 2) +C

Q.19

2 (sin x + x cos &) +C

sec x # cosec x + C

 

 

1 2

x

#

(x # sin x) + C

 

Q.30

# 2 cos x + C

 

Q.35

2

'x

8

 + C

 + C

tan x # tan#1 x + c

#

cos4x

Q.27

x

2

Q.24

#1 1 0 " tan x 3  +C 2 /3 2 tan x # cot x # 3x + C

Q.22

 

2

+C

1 . x3

 + C

# 2  cos

Q.32

ln(2x+1)

 sin xº + C

 

(sin x + cos x) sgn (cos x - sin x) + C x

tan x # x + C

Q.4

sin 2x + C

# (cot x + tan x) + C

Q.7

 + c

 + C   Q.12

#1

3

2

1 1 1 .1 1 # 0 cos 9 x " cos 10 x " cos 11x " cos 12 x 3 + C 10 11 12 /9 2

 + tan x + C Q.9  

"1

a "1

#1

x

1

(tan x + x) + C Q.3

xa

Q.6

 + C

Q.8

Q.20

2

3

3

x

1

Q.2

 + C

#

2 x

x

1 4

 sin#1

4 3

x +C

2

Q.38

4

tan x # sec x + C

 + tan #1 x + C

 –  2tan – 1x   Q.43 #

1 64

cos 8x + C   Q.44

xx + C