Indefinite Integrals MCQ Quiz - Objective Question with Answer for Indefinite Integrals - Download Free PDF

Calculation:

On differentiating both sides, we get

Hence, the correct answer is Option 4.

Calculation:

On differentiating both sides, we get

Hence, the correct answer is Option 3.

Calculation:

On differentiating both sides, we get

Hence, the correct answer is Option 1.

Calculation:

We know that the integral is of the form:

Now, we substitute the given value of f(3) to find the constant C :

⇒ 

We are given that:

⇒ 

Equating the two expressions for f(3):

⇒ 

Since both sides are equal, we conclude that C = 0 .

Thus, the function becomes:

⇒ 

Now, we can calculate f(4):

⇒ 

Thus, the value of f(4) is:

⇒ 

Hence, the correct answer is Option 1.

Calculation:

Let I = 

= 

= 

= 

=  [∵ ]

= 

∴ The value of the integral is .

The correct answer is Option 2.

Concept:

1 + cos 2x = 2cos² x

1 - cos 2x = 2sin² x

Calculation:

I = 

= 

= 

= 

= 

Concept:

Calculation:

I = 

= 

Let 5x = t

Differentiating with respect to x, we get

⇒ 5dx = dt

⇒ dx = 

Now,

I = 

=  + c

=  + c

Concept:

Calculation:

I = 

Let 2x + 3 = t²

Differenating with respect to x, we get

⇒ 2dx = 2tdt

⇒ dx = tdt

Now,

I = 

= 

= 

∵ 2x + 3 = t2

⇒  (2x + 3)^1/2 = t

⇒ (2x + 3)³/2 = t³

⇒ I = 

Concept:

Calculation:

I = 

Let 5x = t

Differentiating with respect to x, we get

⇒ 5dx = dt

⇒ dx = 

Now,

I = 

= 

= 

Concept:

From the Standard integral:

a\)

Calculation:

let t = e^x

dt = e^x dx

From the standard integral:

Put t = ex in the above equation, we get:

Note:

Some important formulas of integration are:

Concept:

Calculation:

Let I =  

Let tan x = t

⇒ sec2x dx = dt

Therefore, the integral becomes.

Re-substitute t = tan x.

Thus,

Concept: 

 =       ....(i)

Calculation:

Let, x² = u

From equation (i)

​ = 

⇒ u +  + + C

Now putting the value of u,

​⇒  = x² +​  +  + C

∴ The required integral is x2 +​  +  + C.

Concept:

Calculation:

Let I = 

Let sin x = t

Now differentiating both sides, we get

⇒ cos x dx = dt

Now,

∴ Option 4 is correct answer

Concept:

• cos 2x = cos² x - sin² x

Calculation:

= 

= 

= 

= 

= - cot x - tan x + C

Concept:

Calculation:

Assume e^-x + 1 = t

Differenatiang with respect to x, we get

⇒ -e^-x dx = dt

∴ e-x dx = -dt