# Test: Definite And Indefinite Integral (Competition Level)

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

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QUESTION: 1

### Let and then

Solution:

We have  QUESTION: 2

### is equal to

Solution: Putting xn = t so that n xn–1 dx = dt  QUESTION: 3

### is equal to

Solution:  QUESTION: 4 Solution:  QUESTION: 5

If then P =

Solution: Comparing it with the given value, we get QUESTION: 6

The value of integral Solution:

put t = 1/x ⇒ dt = -1/x2 as t = π/2 and π QUESTION: 7 Solution:

Put x = 2 cos θ ⇒ dx = - 2 sin θ dθ, then QUESTION: 8

If then

Solution:

Integrate it by parts taking  log (1+ x/2 )as first function  QUESTION: 9

The value of is

Solution: Since sinq is positive in interval (0, π) QUESTION: 10 Solution:  QUESTION: 11 Solution:  QUESTION: 12 Solution: By adding (i) and (ii), we get Now, Put tan2x = t, we get QUESTION: 13 Solution: QUESTION: 14 denotes the greater integer less than or equal to x

Solution:  QUESTION: 15

If [x] denotes the greater integer less than or equal to x, then the value of Solution:  QUESTION: 16

If f(x) = tan x - tan3 x + tan5 x - …… to ∞ with 0 < x < π/4, then Solution:  QUESTION: 17 Solution:

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

QUESTION: 18 Solution: QUESTION: 19 Solution:  QUESTION: 20 Solution:

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

QUESTION: 21 Solution:  QUESTION: 22 Solution:

Here on adding we get QUESTION: 23

If then

Solution:  QUESTION: 24 then

Solution: Differentiating both sides, we get Comparing the coefficient of like terms on both sides, we get QUESTION: 25 Solution: Differentiating both sides, we get Comparing the like powers of x in both sides, we get QUESTION: 26

If then

Solution:  QUESTION: 27 is equal to

Solution:

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

QUESTION: 28 is equal to

Solution: QUESTION: 29 is equal to

Solution:  QUESTION: 30 Solution:

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