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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.
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Competition Level Test: Definite And Indefinite Integral - Question 1

Let   and  then

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

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

Putting xn = t so that n xn–1 dx = dt

Competition Level Test: Definite And Indefinite Integral - Question 3

 is equal to

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

 

Competition Level Test: Definite And Indefinite Integral - Question 4

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

Competition Level Test: Definite And Indefinite Integral - Question 5

If  then P =

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 

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

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

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

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

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

 

Competition Level Test: Definite And Indefinite Integral - Question 11

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

 

Competition Level Test: Definite And Indefinite Integral - Question 12

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

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

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 

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 

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

Competition Level Test: Definite And Indefinite Integral - Question 17

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

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

Competition Level Test: Definite And Indefinite Integral - Question 19

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

Competition Level Test: Definite And Indefinite Integral - Question 20

Detailed Solution for Competition 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

Competition Level Test: Definite And Indefinite Integral - Question 21

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

Competition Level Test: Definite And Indefinite Integral - Question 22

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

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

Competition Level Test: Definite And Indefinite Integral - Question 24

 then

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

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

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

Competition Level Test: Definite And Indefinite Integral - Question 27

 is equal to

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

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

Competition Level Test: Definite And Indefinite Integral - Question 29

 is equal to

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

Competition Level Test: Definite And Indefinite Integral - Question 30

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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