SMR Mathzone Dr.

Ashwin Prakash SV

SMR Mathzone
PYQs on Chapter 7, 8 and 9
Integrals, Applications of Integrals and Differential Equations
65/1/1
MCQ
𝑑
5. If 𝑑𝑥 (𝑓(𝑥)) = log 𝑥, then 𝑓(𝑥) = ?
1 1
(a) − 𝑥 + C (b) 𝑥(log 𝑥 − 1) + 𝐶 (c) 𝑥9 log 𝑥 + 𝑥) + 𝐶 (d) 𝑥 + 𝐶
𝜋/6 𝜋
6. ∫0 sec 2 (𝑥 − 6 ) 𝑑𝑥 = ?
1 1
(a) (b) − (c) √3 (d) −√3
√3 √3

𝑑2 𝑦 𝑑𝑦 3
7. The sum of the order and the degree of the differential equation 𝑑𝑥 2 + (𝑑𝑥 ) = sin 𝑦 is :

(a) 5 (b) 2 (c) 3 (d) 4

15. The general solution of the differential equation 𝑥 𝑑𝑦 − (1 + 𝑥 2 )𝑑𝑥 = 𝑑𝑥 is :
𝑥3 𝑥3
(a) 𝑦 = 2𝑥 + +𝐶 (b) 𝑦 = 2 log 𝑥 + +𝐶
3 3

𝑥2 𝑥2
(c )𝑦 = +𝐶 (d) 𝑦 = 2 log 𝑥 + +𝐶
2 2

SA
𝜋
26. Evaluate: ∫02 log(sin 𝑥) − log (2 cos 𝑥) dx
1
27. Find: ∫ 𝑑𝑥
√𝑥(√𝑥+1)(√𝑥+2)

𝑑𝑦
28. (a) Find the particular solution of the differential equation 𝑑𝑥 + sec 2 𝑥 . 𝑦 = tan 𝑥 . sec 2 𝑥
given that 𝑦(0) = 0.
OR

(b) Solve the differential equation given by 𝑥 𝑑𝑦 − 𝑦 𝑑𝑥 − √𝑥 2 + 𝑦 2 𝑑𝑥 = 0.

𝜋
31. (a) Evaluate: ∫02 𝑒 𝑥 sin 𝑥 𝑑𝑥

OR
1
(b) Find : ∫ (cos(𝑥−𝑎) cos(𝑥−𝑏)) 𝑑𝑥

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SMR Mathzone Dr. Ashwin Prakash SV

LA
35. Find the area of the region bounded by the curves 𝑥 2 = 𝑦, 𝑦 = 𝑥 + 2 and 𝑥 − axis, using
integration.
65/2/1
𝑑
5. If 𝑑𝑥 (𝑓(𝑥)) = 𝑎𝑥 + 𝑏 and 𝑓(0) = 0, then 𝑓(𝑥) =

𝑎𝑥 2 𝑎𝑥 2
(a) a+b (b) + 𝑏𝑥 (c) + 𝑏𝑥 + 𝑐 (d) b
2 2

𝑑𝑦
6. Degree of the differential equation sin 𝑥 + cos (𝑑𝑥 ) = 𝑦 2 =

(a) 2 (b) 1 (c) not defined (d) 0

(1−𝑦 2 )𝑑𝑥
7. The integrating factor of the differential equation + 𝑦𝑥 = 𝑎𝑦. (−1 < 𝑦 < 1) is
𝑑𝑦

1 1 1 1
(a) 𝑦 2−1 (b) (c) 1−𝑦 2 (d)
√𝑦 2 −1 √1−𝑦 2

tan 𝑥−1
8. Anti- derivative of tan 𝑥+1 w.r. to x is
𝜋 𝜋
(a) sec 2 ( 4 − 𝑥) + 𝑐 (b) − sec 2 ( 4 − 𝑥) + 𝑐
𝜋 𝜋
(c) log |sec ( − 𝑥)| + 𝑐 (d) −log |sec ( − 𝑥)| + 𝑐
4 4

SA
𝜋
26. (a) Evaluate ∫04 log(1 + tan 𝑥)𝑑𝑥

OR
𝑑𝑥
(b) Find ∫
√sin3 𝑥 cos(𝑥−𝑎)

−1 𝑥 (1−𝑥+𝑥 2 )
27. Find ∫ 𝑒 cos ( )dx
1+𝑥 2

log √3 1
28. Evaluate ∫log √2 𝑑𝑥
(𝑒 𝑥 +𝑒 −𝑥 )(𝑒 𝑥 −𝑒 −𝑥 )

29. (a) Find the general solution of the differential equation:

(𝑥𝑦 − 𝑥 2 )𝑑𝑦 = 𝑦 2 𝑑𝑥
OR
(b) Find the general solution of the differential equation:
𝑑𝑦
(𝑥 2 + 1) + 2𝑥𝑦 = √𝑥 2 + 4
𝑑𝑥

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SMR Mathzone Dr. Ashwin Prakash SV

LA
34. Using integration, find the area of region bounded by line 𝑦 = √3𝑥, the curve 𝑦 =
√4 − 𝑥 2 and
y axis in first quadrant.

65/3/1
6. ∫ 2𝑥+2 𝑑𝑥 =
(a) 2𝑥+2 + 𝐶 (b) 2𝑥+2 log 2 + 𝐶
(c) 2𝑥+2 / log 2 + 𝐶 (d) 2. 2𝑥 / log 2 + 𝐶
2 𝑐𝑜 2𝑥−1
7. ∫ 𝑑𝑥 =
1+2 sin 𝑥

(a) 𝑥 − 2 cos 𝑥 + 𝐶 (b) 𝑥 + 2 cos 𝑥 + 𝐶

(c) −𝑥 − 2 cos 𝑥 + 𝐶 (d) −𝑥 + 2 cos 𝑥 + C
𝑑𝑥 𝑑𝑦
8. The solution of the differential equation + =0=
𝑥 𝑦

1 1
(a) 𝑥 + 𝑦 = 𝐶 (b) log 𝑥 − log 𝑦 = 𝐶 (c) 𝑥𝑦 = 𝐶 (d) 𝑥 + 𝑦 = 𝐶

𝑑2 𝑦
9. What is the product of the order and degree of the differential equation 𝑑𝑥 2 sin 𝑦 +
𝑑𝑦 3
(𝑑𝑥 ) cos 𝑦 = √𝑦?

(a) 3 (b) 2 (c) 6 (d) not defined

SA
𝑑
26. (a) Find the general solution of the differential equation: 𝑑𝑥 (𝑥𝑦 2 ) = 2𝑦(1 + 𝑥 2 )

OR
𝑑𝑦
(b) Solve the following differential equation: 𝑥𝑒 𝑦/𝑥 − 𝑦 + 𝑥 𝑑𝑥 = 0

3 √4−𝑥
27. Evaluate: ∫1 𝑑𝑥
√ √4−𝑥
𝑥+
𝑒 1
28. Evaluate: ∫1 𝑑𝑥
√4𝑥 2 −(𝑥 log 𝑥)2

cos 𝑥
29. (a) Find: ∫ (sin 3𝑥) 𝑑𝑥

OR

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SMR Mathzone Dr. Ashwin Prakash SV

(b) Find : ∫ (𝑥 2 log(𝑥 2 + 1)) 𝑑𝑥

LA
33. The area of the region bounded by the line 𝑦 = 𝑚𝑥 (𝑚 > 0), the curve 𝑥 2 + 𝑦 2 = 4
𝜋
and the x-axis in the first quadrant is 2 units. Using integration, find the value of m.

65/4/1

8. ∫ 𝑒 5 log 𝑥 dx =
𝑥5 𝑥6
(a) +𝐶 (b) +𝐶 (c) 5𝑥 4 + 𝐶 (d) 6𝑥 5 + 𝐶
5 6
𝑎
9. If ∫0 3𝑥 2 𝑑𝑥 = 8, then the value of a is

(a) 2 (b) 4 (c) 8 (d) 10

𝑑𝑦
10. The integrating factor for solving the differential equation 𝑥 𝑑𝑥 − 𝑦 = 2𝑥 2 is :
1
(a) 𝑒 −𝑦 (b) 𝑒 −𝑥 (c) 𝑥 (d) 𝑥
𝑑𝑦
11. The order and degree of the differential equation (𝑦 ′ )2 + (𝑦 ′ )3 = 𝑥 sin (𝑑𝑥 ) respectively
are:
(a) 2, 2 (b) 1, 3 (c) 2, 3 (d) 2, degree not
defined
SA
𝑥 2 +𝑥+1
26. Find : ∫ (𝑥+1)2 (𝑥+2)
𝑑𝑥
𝜋
(1−sin 2𝑥)
27. (a) Evaluate : ∫𝜋2 𝑒 2𝑥 𝑑𝑥
1−cos 2𝑥
4

OR
2 𝑥2
(b) Evaluate : ∫−2 1+5𝑥 𝑑𝑥
𝑒𝑥
28. (a) Find : ∫ √5−4𝑒 𝑥 −𝑒 2𝑥
𝑑𝑥

OR
𝜋
(b) Evaluate : ∫0 √sin 𝑥 cos5 𝑥 𝑑𝑥
2

𝑑𝑦 𝑥+𝑦
29. (a) Find the particular solution of the differential equation 𝑑𝑥 = , 𝑦(1) = 0
𝑥

OR

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SMR Mathzone Dr. Ashwin Prakash SV

(b) Find the general solution of the differential equation𝑒 𝑥 tan 𝑦 𝑑𝑥 + (1 −

𝑒 𝑥 ) sec 2 𝑦 𝑑𝑦 = 0
LA
33. Using Integration, find the area of the region bounded by the parabola 𝑦 2 = 4𝑎𝑥 and its
latus rectum.
65/5/1
sec 𝑥
10. ∫ dx =
sec 𝑥−tan 𝑥

(a) sec x – tan x + c (b) sec x + tan x + c

(c) tan x – sec x +c (d) –(sec x + tan x) + c
|𝑥−2|
11. ∫ ( 𝑥−2 ) 𝑑𝑠, 𝑥 ≠ 2 =

(a) 1 (b) -1 (c) 2 (d) -2

𝑑 𝑑𝑦 3
12. The sum of the order and degree of the differential equation 𝑑𝑥 (𝑑𝑥 ) is

(a) 2 (b) 3 (c) 5 (d) 0

8 √10−𝑥
20. Assertion (A): ∫2 𝑑𝑥 = 3
√𝑥+√10−𝑥
𝑏 𝑏
Reason:∫𝑎 𝑓(𝑥)𝑑𝑥 = ∫𝑎 𝑓(𝑎 + 𝑏 − 𝑥)𝑑𝑥

VSA
23. Sketch the region bounded by the lines 2𝑥 + 𝑦 = 8, 𝑦 = 2, 𝑦 = 4 and the y-axis. Hence,
obtain its area using integration.
SA
2𝜋 1
28. (a) Evaluate: ∫0 𝑑𝑥
1+𝑒 sin 𝑥

OR
𝑥4
(b) Find: ∫ (𝑥−1)(𝑥 2 +1)
𝑑𝑥

29. Find the area of the following region using integration:

{(𝑥, 𝑦): 𝑦 2 ≤ 2𝑥 & 𝑦 ≥ 𝑥 − 4}
LA
𝜋
33. Evaluate: ∫02 sin 2𝑥 tan−1(sin 𝑥) 𝑑𝑥

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SMR Mathzone Dr. Ashwin Prakash SV

Case Study
38. An equation involving derivatives of the dependent variable with respect to the
independent variables is called a differential equation. A differential equation of the form
𝑑𝑦
= 𝐹(𝑥, 𝑦) is said to be a homogeneous if 𝐹(𝑥, 𝑦) is a homogeneous function of degree 0,
𝑑𝑥
whereas a function 𝐹(𝑥, 𝑦)is a homogeneous function of degree n if 𝐹(𝜆𝑥, 𝜆𝑦) = 𝜆𝑛 𝐹(𝑥, 𝑦).
𝑑𝑦 𝑦
To solve a homogeneous differential equation of the type 𝑑𝑥 = 𝐹(𝑥, 𝑦) = 𝑔 (𝑥 ), we make the
substitution 𝑦 = 𝑣𝑥 and the separate the variables.
Based on the above, answer the following questions:
(I) Show that (𝑥 2 − 𝑦 2 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦 = 0 is a differential equation of the type
𝑑𝑦 𝑦
= 𝑔 (𝑥 ).
𝑑𝑥
(II) Solve the above equation to find its general solution.