Integer Answer Type Questions for JEE: Indefinite Integral

Q.1. Find the maximum area bounded by the curves y² = 4ax, y = ax and y = x / a (1 ≤ a ≤ 2).

Ans. 84

The curves are y² = 4ax and y = ax
At their point of intersection
a²x² = 4ax
⇒ ax = 4, x = 0
x = 4  /  a
⇒ y = 4
i. e.
Similarly for y² = 4ax and y = x / a ,

⇒ x = 4a3
⇒ B(4a³, 4a²)
Area OAB =


∴
= 84

Q.2. Let f be a real valued function satisfying   and  Find the area bounded by the curve y = f(x), the y-axis and the line y = 3. 

Ans. 3

Given,
putting x = y =1, we get  f(1) = 0
Now,
=
⇒ f'(x)  = 3 x ⇒ f(x) = 3 lnx + c
Putting x =1 ⇒ c = 0 ⇒ f(x) = 3lnx = y  (say)
Required Area =
= 3(e - 0) = 3e sq. units.

Q.3.  is equal to

Ans. 0
Given Limit =
=

Q.4. The value of  ,where [.] denotes the greatest integer function, is equal to;

Ans. 12
Let I =
= 1/3 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) = 12.

Q.5. If   , then the value od P + Q + R is________.

Ans. 10
Put x + 7 = u⁴   ⇒ dx = 4u³ du
⇒
Rearranging like terms

⇒ P = 2, Q = 4, R = 4
⇒ P + Q + R = 10.

Q.6. If   ,  then k is equal   s.

Ans. 1
Dividing the numerator and denominator by  x², the   given  integral  becomes

Let
Hence   k = 1.

Q.7. The value of  is ____________.

Ans. 9

Q.8.  If , then  where  k  has the value equal to?

Ans. 9

⇒
⇒
⇒
⇒ k = 9

Q.9. Value of  is equal to?

Ans. 1

⇒
= 1

Q.10. Let  f be a function defined by , where r = 3k, k∈ I then  is equal to? 

Ans. 5

Hence, f(x) is periodic with fundamental time period = 3
 
= 50
⇒
= 5