Integration using Trigonometric Identities MCQ Quiz - Objective Question with Answer for Integration using Trigonometric Identities - Download Free PDF

Concept Used:

tan(x) = sin(x) / cos(x).

Hence, the integral can be rewritten as:

∫ tan(x) dx = ∫ (sin(x) / cos(x)) dx.

We use substitution to simplify this expression.

Calculation:

Let cos(x) = u, then:

du = -sin(x) dx.

Substituting these into the integral:

∫ (sin(x) / cos(x)) dx = ∫ (-1 / u) du.

The integral of -1 / u is:

-ln |u| + C.

Substituting back u = cos(x):

-ln |cos(x)| + C.

∴ The solution to ∫ tan(x) dx is:

-ln(cos(x)) + C.

Hence, the correct answer is Option 2.

Calculation

∵ 

⇒ 

= 

Note : assuming g(x) = 

Comment : In this question we will not get a unique function g(x), but in order to match the answer we will have to assume g(x) = .

Hence option 3 is correct

Calculation

Given integral: ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

⇒ ∫ dx

Let t = 1 + tan x, then dt = sec² x dx

⇒ ∫ dt

⇒ ∫ dt

⇒ ∫ () dt

⇒ 2 ln |t| - t + c

⇒ 2 ln |1 + tan x| - (1 + tan x) + c

⇒ 2 ln |1 + tan x| - tan x - 1 + c

⇒ 2 ln |1 + tan x| - tan x + C (where C = c - 1)

∴ The integral is 2 ln |1 + tan x| - tan x + C

Hence option 4 is correct

Calculation

Let

Let's consider the derivative of :

Therefore,

⇒

∴ The integral is

Hence option 4 is correct

Calculation:

= 

= 

= 

But I = e^xf(x) + C (given)

∴ 

Hence option 3 is correct

Concept:

1 + cos 2x = 2cos² x

1 - cos 2x = 2sin² x

Calculation:

I = 

= 

= 

= 

= 

Concept:

1 - cos 2x = 2 sin² x

1 – sin² x = cos² x

Calculation:

Let I = 

= 2 [tan x – x] + c

= 2 tan x – 2x + c

Concept:

Calculation:

We have,

∫ secn x tan x dx

⇒ ∫ sec ^{n - 1} x (sec x tan x) dx      ----(1)

Let sec x = t

⇒ sec x tan x dx = dt

On substituting these values in equation (1), we get

∫ t ^{n - 1} dt 

Hence, ∫ secn x tan x dx = 

Concept:

sin²x + cos²x = 1

Calculation:

We have  = A sinx + B cosx + C

⇒  = A sinx + B cosx + C

⇒  = A sinx + B cosx + C

⇒  = A sinx + B cosx + C     ----(i)

If 0 , then sinx

⇒ |sinx - cosx| = -sinx + cosx    -----(ii)

Now from (i) and (ii), we get 

⇒  = A sinx + B cosx + C

⇒ cosx + sinx + C = A sinx + B cosx + C

On comparing A = 1, B = 1 and C = 0

Hence, A + B - 2 = 0 is correct.

Concept:

Calculation:

To Find: Integral of sec² x with respect to sec x

Replace sec x = t, we get

Concept:

180° = π radian

Calculation:

As we know that, 180° = π radian

∴ 1° =  radian

So, x° =  radian

Let I = 

Let = t

⇒ dx =  dt

Concept:

Calculation:

Let I = 

= tan x - (-cot x) + c

            (∵ 2 sin x cos x = sin 2x)

= 2 cosec 2x + c

Formula:

Calculation:

Let 

⇒ 

⇒ 

⇒ 

The above integrand is of the form 

Concept:

1 + cos 2x = 2 cos² x

1 – cos² x = sin² x

Calculation:

Let I =     ---- (∵ 1 + cos 2x = 2 cos2 x and 1 – cos2 x = sin2 x)

= 2 [-cot x – x] + c

= -2 cot x – 2x + c

Concept:

Some useful formulas are:

∫cosx dx = sinx + c

∫ secx dx = ln(sec x + tan x) + c

cos2θ = 2cos²θ - 1

= secθ 

Calculation:

Given integration is, 

= 

= 

= 

= 2sinx - ln(sec x + tan x) + C, C = constant of integration