Evaluate using Integration by Parts MCQ Quiz in தமிழ் - Objective Question with Answer for Evaluate using Integration by Parts - இலவச PDF ஐப் பதிவிறக்கவும்

Calculation:

Let

Since the integrand changes sign at x=1, split:

By symmetry,

Consider the antiderivative:

Hence

Evaluating gives

Hence, the correct answer is Option 3.

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.

Calculation

 = 2

Calculation:

Given that,

 dx

Putting x² = t

differentiating w.r.t "t" we get,

⇒ 2x dx = dt ⇒ x dx = 

x = 0, t = 0, x = 1, t = 1

Now, 

= 

The correct answer is option "1'

Explanation:

Concept: Integration by part

If f and g are two functions, then ∫fg = f∫g - ∫{f'∫g}

If ∫ f(x)dx = F(x), then f(x) = F(b)-F(a)

Let 

We have f = x³ and g = sinx

∴ 

⇒ 

On applying limits of integration,

∴ I = (3(π /2)² - 6)
⇒ 

Concept:

Integration By Parts of Definite Integrals formula: 

Calculation:

Given:

Here, u= , v = 

So apply by Parts:

 --------(Since u' = )

 -------( Since )

Put the limits we get,

Now,

 ---(1)

 ----(2)

Subtract (1) from (2)

(20)I10 = 10I9 + 9I8

Comparing with (20)I10 = αI9 + βI8

we get

α = 10 and β = 9

So  α - β = 1