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JEE Advanced Level Test: Definite and Indefinite Integral - JEE MCQ


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30 Questions MCQ Test Mathematics (Maths) Class 12 - JEE Advanced Level Test: Definite and Indefinite Integral

JEE Advanced Level Test: Definite and Indefinite Integral for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The JEE Advanced Level Test: Definite and Indefinite Integral questions and answers have been prepared according to the JEE exam syllabus.The JEE Advanced Level Test: Definite and Indefinite Integral MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for JEE Advanced Level Test: Definite and Indefinite Integral below.
Solutions of JEE Advanced Level Test: Definite and Indefinite Integral questions in English are available as part of our Mathematics (Maths) Class 12 for JEE & JEE Advanced 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 JEE Advanced 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
JEE Advanced Level Test: Definite and Indefinite Integral - Question 1

Let   and  then

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 1

We have 

JEE Advanced Level Test: Definite and Indefinite Integral - Question 2

 is equal to

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 2

Putting xn = t so that n xn–1 dx = dt

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JEE Advanced Level Test: Definite and Indefinite Integral - Question 3

 is equal to

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 3

 

JEE Advanced Level Test: Definite and Indefinite Integral - Question 4

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 4

JEE Advanced Level Test: Definite and Indefinite Integral - Question 5

If  then P =

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 5

Comparing it with the given value, we get

JEE Advanced Level Test: Definite and Indefinite Integral - Question 6

The value of integral 

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 6

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

JEE Advanced Level Test: Definite and Indefinite Integral - Question 7

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 7

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

JEE Advanced Level Test: Definite and Indefinite Integral - Question 8

If   then

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 8

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

JEE Advanced Level Test: Definite and Indefinite Integral - Question 9

The value of  is

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 9

Since sinq is positive in interval (0, π)

JEE Advanced Level Test: Definite and Indefinite Integral - Question 10

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 10

 

JEE Advanced Level Test: Definite and Indefinite Integral - Question 11

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 11

 

JEE Advanced Level Test: Definite and Indefinite Integral - Question 12

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 12

 

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

Now, Put tan2x = t, we get

JEE Advanced Level Test: Definite and Indefinite Integral - Question 13

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JEE Advanced Level Test: Definite and Indefinite Integral - Question 14

denotes the greater integer less than or equal to x

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 14

JEE Advanced Level Test: Definite and Indefinite Integral - Question 15

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

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 15

JEE Advanced Level Test: Definite and Indefinite Integral - Question 16

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

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 16

JEE Advanced Level Test: Definite and Indefinite Integral - Question 17

Detailed Solution for JEE Advanced 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

JEE Advanced Level Test: Definite and Indefinite Integral - Question 18

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 18

JEE Advanced Level Test: Definite and Indefinite Integral - Question 19

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 19

JEE Advanced Level Test: Definite and Indefinite Integral - Question 20

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 20

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

JEE Advanced Level Test: Definite and Indefinite Integral - Question 21

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JEE Advanced Level Test: Definite and Indefinite Integral - Question 22

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 22

Here  on adding we get 

JEE Advanced Level Test: Definite and Indefinite Integral - Question 23

If then

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 23

JEE Advanced Level Test: Definite and Indefinite Integral - Question 24

 then

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 24

Differentiating both sides, we get

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

JEE Advanced Level Test: Definite and Indefinite Integral - Question 25

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 25

Differentiating both sides, we get

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

 

JEE Advanced Level Test: Definite and Indefinite Integral - Question 26

If  then

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 26

JEE Advanced Level Test: Definite and Indefinite Integral - Question 27

 is equal to

Detailed Solution for JEE Advanced 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

JEE Advanced Level Test: Definite and Indefinite Integral - Question 28

 is equal to

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 28

JEE Advanced Level Test: Definite and Indefinite Integral - Question 29

 is equal to

Detailed Solution for JEE Advanced Level Test: Definite and Indefinite Integral - Question 29

JEE Advanced Level Test: Definite and Indefinite Integral - Question 30

Detailed Solution for JEE Advanced 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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