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

If ">, then

And if x = \(\dfrac{\pi}{3}\), then 0\)

Concept:

The given integral is:

This represents a double integral over the upper half of the unit circle centered at the origin. So, converting it into polar coordinates simplifies the integration process.

Let and

The region of integration becomes a semicircle:

Calculation:

In polar coordinates,

So, the integral becomes:

Correct Option:

The correct answer is: 3) π/5

Explanation:

We are given the integral:

Integral: I =  

Step 1: Substitution

Let u = a + bx², then du = 2bx dx.

Hence, x dx = du / (2b).

After substitution, the limits change: when x = 0, u = a, and

when x = 1, u = a + b.

The integral becomes:

I = 1 / (2b) ∫_a^a+b (2a - u) / u² du

Step 2: Decompose the Integral

I = 1 / (2b) [ 2a ∫_a^a+b 1/u² du - ∫_a^a+b 1/u du ]

Each part is solved as follows:

• ∫_a^a+b 1/u² du = [ -1/u ]_a^a+b = 1/a - 1/(a+b)
• ∫_a^a+b 1/u du = ln(u)_a^a+b = ln(a+b) - ln(a)

Thus, the integral becomes:

I = 1 / (2b) [ 2a (1/a - 1/(a+b)) - (ln(a+b) - ln(a)) ]

Final Result:

The value of the integral is: I = 1 / (a + b)

Calculation

Let x = sin²θ dx = 2sinθcosθdθ

⇒ 

⇒ I (9, 14) + I (10, 13) = 

⇒ 

⇒ I (9, 13) 

Hence option 4 is correct

Calculation

= 

= -x³ cos x + 3x² sin x + 6x cos x - 6 sin x + c

So g(x) = -x3 cos x + 3x2 sin x + 6x cos x - 6 sin x 

g'(x) = 3x² cos x + x³ sin x + 6cos x - 6cos x

So α + β – γ = 55

Hence option 1 is correct

Concept:

• sin² x + cos² x = 1
• sin 2A = 2 sin A cos A
• ∫ cos x = sin x

• ∫ sin x = -cos x

Calculation:

Hence, 

Concept:

Calculation:

Let, 

        ----(  = 1)

       -----

Let 

Differentiating with respect to x, we get

Limits become for x = 0, t = 0

for x = 1, t = 1/4 = 0.25

Hence, option (4) is correct.

Concept:

1 + tan² x = sec² x

Calculation:

             (∵ 1 + tan² x = sec² x)

Let tanx = t

Differentiating with respect to x, we get

⇒ sec2 x dx = dt

∴ The value of the integral is 1/2.

Concept:

Calculation:

 Given: 

                  (1 = )

= 

= 

= 

= 0

Concept:

Definite Integral:

If ∫ f(x) dx = g(x) + C, then  = g(b) - g(a).

Trigonometric Identities:

cos 2x = 2 cos² x - 1

Calculation:

Let I = 

⇒ I = 

⇒ I = 

⇒ I = 

⇒ I = 

⇒ I = .

Concept:

Calculation:

Let, 

Hence, option (4) is correct.

Concept:

sin² x + cos² x = 1

Calculation:

I = 

= 

= 

Let, sin x = u

cos x dx = du

= 

= 

= 1 - 0 - 

= 

Formula used:

tan(π - θ) = - tan θ  

Calculation:

Let

I =         ---(1)

According to the formula used

I = 

⇒ I = - 

From equation (1)

⇒ I = -I

⇒ 2I = 0

⇒ I = 0

∴ The value of the integral  is 0.

Concept:

Definite Integral:

If ∫ f(x) dx = g(x) + C, then  = g(b) - g(a).

Calculation:

Let I = 

⇒ I = 

⇒ I = 

⇒ I = 0.

Concept:

cot² x = cosec² x - 1 

Calculation:

I =  

I =  

I =  

I =  

I =  

I =            ...(1)

Let , sin x = t 

Differentiate both side  w.r.t  x , 

cos x dx = dt   

If , x = 0 , then t = 0 

x = π /4 , then t =  

substitute above values in eq. (1) 

I =   

I =  

I =   

I = ∞  

∴ The correct option is 4).