Evaluate using Integration by Parts MCQ Quiz - Objective Question with Answer for Evaluate using Integration by Parts - Download Free PDF

Explanation:

Given:

f(x) =|x² – x – 2|

= {x²- x - 2; x ∈ (-∞, -1) ∪ (2, ∞) 

      - (x2 -x -2 ; x ∈[ -1,2]

Let I = 

= 

= 

= 

=  = 3

∴ Option (b) is correct.

Explanation:

Given:

f(x) =|x² – x – 2|.

= {x² – x – 2; x ∈ (-∞ –1) ∪  (2,∞ ) – (x² – x – 2); x ∈ [–1, 2]

Let I = - 

= - 

= 

∴ Option (d) is correct

Calculation

 = 2

Explanation -

Given the equation:

 ..... (i)

Substitute x with 1/x :

...... (ii)   

We now have two equations:

Let: f(x) = a and f(1/x) = b

The system of equations becomes:

3a + b = 1/x + 1
3b + a = x + 1

Multiply the first equation by 3:

9a + 3b = 3/x + 3

Subtract the second equation from the modified first equation:

9a + 3b - (3b + a) = 3/x + 3 - (x + 1)

⇒ 8a = 3/x + 3 - x - 1

⇒ 8a = 3/x - x + 2

⇒ a =

Therefore: f(x) =   

Thus, the function f(x) = 

Now  

= 

= 

= 

= 

= 

= 

= 

= ln 8 + ln√e

= ln 8√e

Hence Option (1) is Correct.

Explanation -

Given the equation:

 ..... (i)

Substitute x with 1/x :

...... (ii)   

We now have two equations:

Let: f(x) = a and f(1/x) = b

The system of equations becomes:

3a + b = 1/x + 1
3b + a = x + 1

Multiply the first equation by 3:

9a + 3b = 3/x + 3

Subtract the second equation from the modified first equation:

9a + 3b - (3b + a) = 3/x + 3 - (x + 1)

⇒ 8a = 3/x + 3 - x - 1

⇒ 8a = 3/x - x + 2

⇒ a =

Therefore: f(x) =   

Thus, the function f(x) =   

Concept:

Integration by parts: Integration by parts is a method to find integrals of products

The formula for integrating by parts is given by;

⇒ ∫ uv dx = u(x) ∫ v(x) dx - ∫ [u'(x) ∫ v(x) dx] dx

Where u is the function u(x) and v is the function v(x), and is chosen by following ILATE rule.

ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Calculation:

Given:

Let I = 

Applying by parts rule, we get

I 

I = (e - 0) - (e - 1)

I = 1

Hence the required value of integration is 1.

Concept:

Modulus Function:

A modulus function is a function which gives the absolute value of a number or variable.

Formula Used:

Calculation:

We have,

⇒ I = 

⇒ 1 - x

⇒ I = 

⇒ I = 

⇒ I =  - 

⇒ I = 

⇒ I = 

⇒ I = 1

∴ The value of  is 1.

Concept:

Integration by parts: Integration by parts is a method to find integrals of products

• The formula for integrating by parts is given by,
• ∫u v dx = u∫v dx −∫u' (∫v dx) dx

Where u is the function u(x) and v is the function v(x)

ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Calculation:

Let I = 

Apply by parts rule, we get

= π3 - 6π

Hence, option (1) is correct.

Concept:

1. Integration by parts: Integration by parts is a method to find integrals of products

The formula for integrating by parts is given by;

 , Where u is the function u(x) and v is the function v(x)

2. ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Calculation:

Let I = 

Apply by parts rule,

= 

Concept:

Integration by parts: Integration by parts is a method to find integrals of products

The formula for integrating by parts is given by;

Where u is the function u(x) and v is the function v(x)

ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Calculation:

First we will calculate the integration without the limits.

Let us consider  .

On differentiating u on both sides we get,

  .

Therefore, we integrate the given function as follows:

Now as the given integral is definite we will remove the constant of integration and put the limits.

Therefore, .

Concept:

Calculation:

Given:

Applying limits,

= [1 × e¹ – e¹] – [0 × e⁰ – e⁰]

= [e¹ – e¹] – [0 - 1]

= 1

Explanation:

Given:

f(x) =|x² – x – 2|.

= {x² – x – 2; x ∈ (-∞ –1) ∪  (2,∞ ) – (x² – x – 2); x ∈ [–1, 2]

Let I = - 

= - 

= 

∴ Option (d) is correct

Explanation:

Given:

f(x) =|x² – x – 2|

= {x²- x - 2; x ∈ (-∞, -1) ∪ (2, ∞) 

      - (x2 -x -2 ; x ∈[ -1,2]

Let I = 

= 

= 

= 

=  = 3

∴ Option (b) is correct.

Concept:

Integration by parts: Integration by parts is a method to find integrals of products

The formula for integrating by parts is given by;

⇒ ∫ uv dx = u(x) ∫ v(x) dx - ∫ [u'(x) ∫ v(x) dx] dx

Where u is the function u(x) and v is the function v(x), and is chosen by following ILATE rule.

ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Calculation:

Given:

Let I = 

Applying by parts rule, we get

I 

I = (e - 0) - (e - 1)

I = 1

Hence the required value of integration is 1.