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QUESTION: 1

Let and then

Solution:

We have

QUESTION: 2

is equal to

Solution:

Putting x^{n} = t so that n x^{n–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 tan^{2}x = 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/x^{2}

Thus, ∫(1 to 2)e^{x}(1/x - 1/x^{2})dx

= [e^{x}/x](1 to 2) + c

= e^{2}/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 = (sec^{2} x)/(tan x) dx

=> (1/cos^2x) * (cosx /sinx) dx = dt

dt = dx/(cosx sinx)

I = ∫t dt

= [t^{2}]/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

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