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Competition Level Test: Definite And Indefinite Integral

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30 Questions MCQ Test Mathematics (Maths) Class 12 | Competition Level Test: Definite And Indefinite Integral

Competition Level Test: Definite And Indefinite Integral for JEE 2023 is part of Mathematics (Maths) Class 12 preparation. The Competition Level Test: Definite And Indefinite Integral questions and answers have been prepared according to the JEE exam syllabus.The Competition Level Test: Definite And Indefinite Integral MCQs are made for JEE 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Competition Level Test: Definite And Indefinite Integral below.

Solutions of Competition Level Test: Definite And Indefinite Integral questions in English are available as part of our Mathematics (Maths) Class 12 for JEE & Competition Level Test: Definite And Indefinite Integral solutions in Hindi for Mathematics (Maths) Class 12 course. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free. Attempt Competition Level Test: Definite And Indefinite Integral | 30 questions in 60 minutes | Mock test for JEE preparation | Free important questions MCQ to study Mathematics (Maths) Class 12 for JEE Exam | Download free PDF with solutions

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Competition Level Test: Definite And Indefinite Integral - Question 1

Let and then

A.
I₁ > I₂
B.
I₂ > I₁
C.
I₁ = I₂
D.
I₁ > 2I₂

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 1

We have

Competition Level Test: Definite And Indefinite Integral - Question 2

is equal to

A.
B.
C.
D.

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 2

Putting xⁿ = t so that n x^n–1 dx = dt

Competition Level Test: Definite And Indefinite Integral - Question 3

is equal to

A.
B.
C.
D.

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 3

Competition Level Test: Definite And Indefinite Integral - Question 4

A.
π/2 - 2log√2
B.
π/2 + 2log√2
C.
π/4 - 2log√2
D.
π/2 + 2log√2

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 4

Competition Level Test: Definite And Indefinite Integral - Question 5

If then P =

A.
1/3
B.
1/4
C.
1/2
D.
1/5

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 5

Comparing it with the given value, we get

Competition Level Test: Definite And Indefinite Integral - Question 6

The value of integral

A.
2
B.
- 1
C.
0
D.
1

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 6

put t = 1/x ⇒ dt = -1/x2 as t = π/2 and π

Competition Level Test: Definite And Indefinite Integral - Question 7

A.
π + 2
B.
π + 3/2
C.
π + 1
D.
π + 5/2

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 7

Put x = 2 cos θ ⇒ dx = - 2 sin θ dθ, then

Competition Level Test: Definite And Indefinite Integral - Question 8

If then

A.
a = 3/2,b=3/2
B.
a = 3/4,b= - 3/4
C.
a = 3/4,b=3/2
D.
a = b

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 8

Integrate it by parts taking log (1+ x/2 )as first function

Competition Level Test: Definite And Indefinite Integral - Question 9

The value of is

A.
0
B.
3/8
C.
4/3
D.
3/4

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 9

Since sinq is positive in interval (0, π)

Competition Level Test: Definite And Indefinite Integral - Question 10

A.
B.
C.
D.

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 10

Competition Level Test: Definite And Indefinite Integral - Question 11

A.
- (π/2 ) log 2
B.
π log 1/2
C.
- π log 1/2
D.
1/2 log2

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 11

Competition Level Test: Definite And Indefinite Integral - Question 12

A.
0
B.
π/8
C.
π²/8
D.
π²/16

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 12

By adding (i) and (ii), we get

Now, Put tan²x = t, we get

Competition Level Test: Definite And Indefinite Integral - Question 13

A.
0
B.
2 ( √2 - 1)
C.
√2 -1
D.
2 ( √2 + 1)

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 13

Competition Level Test: Definite And Indefinite Integral - Question 14

denotes the greater integer less than or equal to x

A.
log 2
B.
log 3
C.
log 5
D.
log 6

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 14

Competition Level Test: Definite And Indefinite Integral - Question 15

If [x] denotes the greater integer less than or equal to x, then the value of

A.
1
B.
2
C.
4
D.
8

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 15

Competition Level Test: Definite And Indefinite Integral - Question 16

If f(x) = tan x - tan3 x + tan5 x - …… to ∞ with 0 < x < π/4, then

A.
1
B.
0
C.
1/4
D.
½

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 16

Competition Level Test: Definite And Indefinite Integral - Question 17

A.
π/2
B.
0
C.
π/4
D.
π/3

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 17

I = ∫0 π2 log(tan x).dx
I = ∫0 π2 log(cot x).dx
Adding both the equations, we get
2I = ∫0 π2 log(tanx) + log(cot x) dx
2I = ∫0 π2 log(1).dx
= 0

Competition Level Test: Definite And Indefinite Integral - Question 18

A.
π/4
B.
π/2
C.
- π/4
D.
- π/2

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 18

Competition Level Test: Definite And Indefinite Integral - Question 19

A.
A = –1/2
B.
A = –1/8
C.
A = –1/4
D.
A= –1/6

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 19

Competition Level Test: Definite And Indefinite Integral - Question 20

A.
e²/2 + e
B.
e - e²/2
C.
e²/2 - e
D.
- e²/2 - e

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 20

f’(x) = -1/x²
Thus, ∫(1 to 2)e^x(1/x - 1/x²)dx
= [e^x/x](1 to 2) + c
= e²/2 - e

Competition Level Test: Definite And Indefinite Integral - Question 21

A.
-2 coseca ( cosa - tan x sin a )^{1/ 2} + C
B.
-2 ( cos a + cot x sin a )^{1/ 2} + C
C.
-2coseca ( cos a + cot x sin a )^{1/ 2} + C
D.
-2coseca (sin a + cot x cos a )^{1/ 2} + C

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 21

Competition Level Test: Definite And Indefinite Integral - Question 22

A.
B.
C.
D.

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 22

Here on adding we get

Competition Level Test: Definite And Indefinite Integral - Question 23

If then

A.
A = 1/4,B = 1/5
B.
A = 1/8,B = -1/5
C.
A = -1/8,B = -1/5
D.
A = -1/4,B = -1/5

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 23

Competition Level Test: Definite And Indefinite Integral - Question 24

then

A.
a = - 1/8,b=7/8
B.
a = 1/8,b=7/8
C.
a = - 1/8,b= - 7/8
D.
a = 1/8,b= - 7/8

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 24

Differentiating both sides, we get

Comparing the coefficient of like terms on both sides, we get

Competition Level Test: Definite And Indefinite Integral - Question 25

A.
a = - 1/10,b = - 2/5
B.
a = 1/10,b = - 2/5
C.
a = - 1/10,b = 2/5
D.
a = 1/10,b = 2/5

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 25

Differentiating both sides, we get

Comparing the like powers of x in both sides, we get

Competition Level Test: Definite And Indefinite Integral - Question 26

If then

A.
a = - 1,b = 1/3
B.
a = - 3,b = 2/3
C.
a = - 2,b = 4/3
D.
a = - 2,b = 2/3

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 26

Competition Level Test: Definite And Indefinite Integral - Question 27

is equal to

A.
B.
C.
D.
None of these

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 27

t = ln(tan x)
dt = (sec² x)/(tan x) dx
=> (1/cos^2x) * (cosx /sinx) dx = dt
dt = dx/(cosx sinx)
I = ∫t dt
= [t²]/2 + c
= 1/2[ln(tanx)]² + c

Competition Level Test: Definite And Indefinite Integral - Question 28

is equal to

A.
B.
C.
D.

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 28

Competition Level Test: Definite And Indefinite Integral - Question 29

is equal to

A.
B.
C.
D.

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 29

Competition Level Test: Definite And Indefinite Integral - Question 30

A.
log 4/3
B.
log 1/3
C.
log 3/4
D.
log 2/3

Detailed Solution for Competition Level Test: Definite And Indefinite Integral - Question 30

ut sin x = t Þ cos x dx = dt, so that reduced integral is

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