DPP for JEE: Daily Practice Problems- Integrals (Solutions)

# Integrals Practice Questions - DPP for JEE

``` Page 1

1. (b) I =
=
=
=
=
=
2. (c) I
n
=  =
= (n – 1)
Page 2

1. (b) I =
=
=
=
=
=
2. (c) I
n
=  =
= (n – 1)
= (n – 1)
? I
n
= (n – 1) I
n – 2
– (n – 1) I
n
? n I
n
= (n – 1) I
n – 2
? I
n
=  I
n – 2
? n (I
n – 2
– I
n
) = I
n – 2
Also,    I
n
: I
n – 2
= (n – 1) : n and I
n – 2
> I
n
.
3. (a) ;
= f (sinx) . cosx – f (2x) × 2
=cos(sin
3
x) . cosx – cos(2x)
3
× 2
= cos(sin
3
x) . cosx – 2cos(8x
3
)
4. (d) Put x = 2a – t
so that dx = – dt
when x = a, t = a and when x = 2a, t = 0
5. (d)
?
?  tan (ln x) tan   tan (ln 2) = tan (ln x) – tan  – tan (ln
2)
?
= ln sec (ln x) – ln sec  – tan (ln 2) ln x
Page 3

1. (b) I =
=
=
=
=
=
2. (c) I
n
=  =
= (n – 1)
= (n – 1)
? I
n
= (n – 1) I
n – 2
– (n – 1) I
n
? n I
n
= (n – 1) I
n – 2
? I
n
=  I
n – 2
? n (I
n – 2
– I
n
) = I
n – 2
Also,    I
n
: I
n – 2
= (n – 1) : n and I
n – 2
> I
n
.
3. (a) ;
= f (sinx) . cosx – f (2x) × 2
=cos(sin
3
x) . cosx – cos(2x)
3
× 2
= cos(sin
3
x) . cosx – 2cos(8x
3
)
4. (d) Put x = 2a – t
so that dx = – dt
when x = a, t = a and when x = 2a, t = 0
5. (d)
?
?  tan (ln x) tan   tan (ln 2) = tan (ln x) – tan  – tan (ln
2)
?
= ln sec (ln x) – ln sec  – tan (ln 2) ln x
= ln
6. (c) Let I =
put x =
=
=
=
= =
Put sin ? = t ? cos ? d? = dt
= = =
=
=
=
= =
Page 4

1. (b) I =
=
=
=
=
=
2. (c) I
n
=  =
= (n – 1)
= (n – 1)
? I
n
= (n – 1) I
n – 2
– (n – 1) I
n
? n I
n
= (n – 1) I
n – 2
? I
n
=  I
n – 2
? n (I
n – 2
– I
n
) = I
n – 2
Also,    I
n
: I
n – 2
= (n – 1) : n and I
n – 2
> I
n
.
3. (a) ;
= f (sinx) . cosx – f (2x) × 2
=cos(sin
3
x) . cosx – cos(2x)
3
× 2
= cos(sin
3
x) . cosx – 2cos(8x
3
)
4. (d) Put x = 2a – t
so that dx = – dt
when x = a, t = a and when x = 2a, t = 0
5. (d)
?
?  tan (ln x) tan   tan (ln 2) = tan (ln x) – tan  – tan (ln
2)
?
= ln sec (ln x) – ln sec  – tan (ln 2) ln x
= ln
6. (c) Let I =
put x =
=
=
=
= =
Put sin ? = t ? cos ? d? = dt
= = =
=
=
=
= =
=
7. (a) I =
Let x
2
= t ? 2x dx = dt
Also, when x = , t = ln2
when x = , t = ln3
? I = ...(1)
Using
We get
I = ...(2)
Adding values of I in equations (1) and (2)
2 I =
? I =
8. (c)
=
[Using identity sin
2
A – sin
2
B = sin (A+B) sin (A – B)]
Page 5

1. (b) I =
=
=
=
=
=
2. (c) I
n
=  =
= (n – 1)
= (n – 1)
? I
n
= (n – 1) I
n – 2
– (n – 1) I
n
? n I
n
= (n – 1) I
n – 2
? I
n
=  I
n – 2
? n (I
n – 2
– I
n
) = I
n – 2
Also,    I
n
: I
n – 2
= (n – 1) : n and I
n – 2
> I
n
.
3. (a) ;
= f (sinx) . cosx – f (2x) × 2
=cos(sin
3
x) . cosx – cos(2x)
3
× 2
= cos(sin
3
x) . cosx – 2cos(8x
3
)
4. (d) Put x = 2a – t
so that dx = – dt
when x = a, t = a and when x = 2a, t = 0
5. (d)
?
?  tan (ln x) tan   tan (ln 2) = tan (ln x) – tan  – tan (ln
2)
?
= ln sec (ln x) – ln sec  – tan (ln 2) ln x
= ln
6. (c) Let I =
put x =
=
=
=
= =
Put sin ? = t ? cos ? d? = dt
= = =
=
=
=
= =
=
7. (a) I =
Let x
2
= t ? 2x dx = dt
Also, when x = , t = ln2
when x = , t = ln3
? I = ...(1)
Using
We get
I = ...(2)
Adding values of I in equations (1) and (2)
2 I =
? I =
8. (c)
=
[Using identity sin
2
A – sin
2
B = sin (A+B) sin (A – B)]
=
=
..........etc.
......... form an H.P.
9. (a) We have,
=
Now,
? J < 2
10. (a) We have, if

Again if
```

174 docs

## DPP: Daily Practice Problems for JEE Main & Advanced

174 docs

### Up next

 Explore Courses for JEE exam
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;