Properties of Definite Integrals MCQ Quiz in मल्याळम - Objective Question with Answer for Properties of Definite Integrals - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

Some useful formulas are:

tan^-11 = 

tan^-10 = 0

Calculation:

Let, cosx = z

∴ - sinx dx = dz 

Substituting the values we get,

= 

= 

= tan^-11 - tan^-10

= 

Concept:

For an odd function f(x): .

Calculation:

We observe that ax⁵ and bx³ are odd functions of x, 

∵ a(-x)5 = -ax⁵ and b(-x)3 = -bx3.

∴  and .

The value of  depends on the value of c.

In fact, 

.

= c [2 - (-2)]

= 4c

Additional Information

A function f(x) is:

• Even, if f(-x) = f(x). And .
• Odd, if f(-x) = -f(x). And .

Concept:

Definite Integral properties:

Trigonometry Formula:

tan ( -x ) = - tan x 

Calculation:

Let , I =                  ....(1)

Now using property, 

I = 

I =                     .... (ii)

Adding equation (i) and (ii), we get

2I =   

2I =  

2I =  

2I =  

2I = π

∴  I = π /2  

The correct option is 3. 

Concept:

Integral Properties:

Calculation:

Given: f(x) = f(a – x) & g(x) + g(a – x) = 2

To find: 

      …. (1)

Using integral property, 

Using given, f(x) = f(a – x)

      …. (2)

Adding equation (1) & (2)

Using given, g(x) + g(a – x) = 2

Concept:

Some useful formulas are:

Calculation:

Let, I = 

Since,  , so

 I = 

∴ I = 

∴ I = 

∴ I =  - 

∴  I =  - I

∴ 2I = 

∴ I = π² 

Concept: 

The rules for integrating even and odd functions ,

If the function is even or odd and the interval is [-a, a], we can apply these rules:

• When f(x) is even ⇔ 
• When f(x) is odd ⇔  

​Calculation: 

Let f (x) = x⁵ sin⁴ x 

f (-x) = (-x)⁵ sin⁴ (-x) = -x⁵ (-sin x )⁴ = -x⁵ sin⁴ x .

⇒ f(-x) = - f(x) 

So, f(x) is an odd function . 

We knoiw that, when f(x) is odd ⇔  

Hence ,  = 0 .  

The correct option is 4 .

Calculation:

Given:           .... (1)

Now,

        (∵ log 

                            (∵ log 1 = 0)

From equation (1), we get

= 

Concept:

Property of definite integrals: 

Calculation:

I =          …. (1)

Using the property of definite integrals: 

I 

I        .... (2)

Adding equations (1) and (2), we get

∴ I = 

Concept:

Odd and even function:

If a function f(x) such that f(-x) = f(x) then f(x) is an even function and if f(-x) = -f(x) then f(x) is an odd function.

Integration of odd and even function:

The following two cases hold for a differentiable function f(x).

If f(x) is an even function then .

If f(x) is an odd function then .

Calculation:

Let , put -x instead of x then we get .

Therefore, f(-x) = f(x) implies that f(x) is an even function.

Thus, the integral is given by,

Hence, the value of the given integral is 2.

Concept:

Calculation:

∴

Hence, option (4) is correct.