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