JEE Mathematics/ Lakshmikanta Satapathy/ Questions on Indefinite Integration Part 13, taken from previous Board Papers solved by the method of substitution using standard integrals
2. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
Answer-1
sec 1.I x dx
(1 cos )(1 cos )
.
cos (1 cos )
x x
dx
x x
1 cos
.
cos
x
dx
x
2
sin
.
cos cos
x
dx
x x
1
1 .
cos
dx
x
3. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
cos sin . sin .Put x t x dx dt x dx dt Substitution :
2
2 1 1
4 4
dt dt
I
t t t t
2 2
1 1 1
log
2 2 2
t t C
21
log
2
t t t C
2 2
1 1
2 2
dt
t
2 2
2 2
log
dx
using x x a
x a
2
[ ]
1
log cos cos cos
2
x x x C Ans
4. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
Answer-2
Substitution : sin cos .Put x t x dx dt
2
2sin2 cos
.
6 cos 4sin
x x
I dx
x x
2
(4sin 1)cos
.
sin 4sin 5
x x
dx
x x
2
(4 1)
.
4 5
t
I dt
t t
1 22 2
2 4
2 . 7 ( )
4 5 4 5
t dt
dt I I say
t t t t
2 2
sin2 2sin cos
cos 1 sin
x x x
x x
2
4sin cos cos
.
6 (1 sin ) 4sin
x x x
dx
x x
2 2
4 8 7 2(2 4) 7
. .
4 5 4 5
t t
dt dt
t t t t
5. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
2 1
1 2 2log 4 5 7tan ( 2)I I I t t t C
2 1
2log sin 4sin 5 7tan (sin 2) [ ]Anx x x C s
1 2
2 4
2 .
4 5
t
I dt
t t
2 2 2
7 7
4 5 4 4 1
dt dt
And I
t t t t
2
4 5 (2 4)Put t t u t dt du
2
1 2 2log 2log 4 5
du
I u t t
u
1
2 2
7 7tan ( 2)
( 2) 1
dt
t
t
1
2 2
1
tan
dx x
using
x a a a
6. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
Answer-3
2
2
.
2 1
x
I dx
x x
1 22 2
1 2 2
.
2 2 1 2 1
x dx
dx I I
x x x x
2
1 (2 2) 2
.
2 2 1
x
dx
x x
2
1 2 4
.
2 2 1
x
dx
x x
1 2
1 2 2
.
2 2 1
x
I dx
x x
2
2 1 (2 2).Put x x t x dx dt
2
1 2 1
2
dt
I t x x
t
7. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
2 2 2
2 1 2 1 2
dx dx
And I
x x x x
2 2
2 1 log ( 1) 2 [ ]1x x x x C Ansx
2
2
log ( 1) ( 1) 2x x 2 2
2 2
log
dx
using x x a
x a
2
log ( 1) 2 1x x x
1 2I I I
2
2
( 1) 2
dx
x
8. Physics Helpline
L K Satapathy
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