Evaluate using Special Integral Forms MCQ Quiz in বাংলা - Objective Question with Answer for Evaluate using Special Integral Forms - বিনামূল্যে ডাউনলোড করুন [PDF]
Last updated on Apr 9, 2025
Latest Evaluate using Special Integral Forms MCQ Objective Questions
Top Evaluate using Special Integral Forms MCQ Objective Questions
Evaluate using Special Integral Forms Question 1:
The value of
Answer (Detailed Solution Below)
Evaluate using Special Integral Forms Question 1 Detailed Solution
Calculation:
We are given the function:
I = ∫-11 |tan-1(x)| dx
Step 1: Split the integral based on the absolute value:
I = ∫-10 -tan-1(x) dx + ∫01 tan-1(x) dx
Step 2: Use the standard result for the integral of tan-1(x):
∫ tan-1(x) dx = x tan-1(x) - 1/2 ln(1 + x2)
Step 3: Compute each integral:
For ∫-10 -tan-1(x) dx, we get:
- [ x tan-1(x) - 1/2 ln(1 + x2) ]-10
At x = 0, the expression becomes 0.
At x = -1, we get:
- [-1 × (-π/4) - 1/2 ln(1 + 1)] = π/4 - 1/2 ln(2)
Thus, the value of the first integral is:
π/4 - 1/2 ln(2)
For ∫01 tan-1(x) dx, we use the same result:
[ x tan-1(x) - 1/2 ln(1 + x2) ]01
At x = 1, the expression becomes:
1 × (π/4) - 1/2 ln(2) = π/4 - 1/2 ln(2)
At x = 0, the expression is 0.
Thus, the value of the second integral is:
π/4 - 1/2 ln(2)
Step 4: Add the results of the two integrals:
I = (π/4 - 1/2 ln(2)) + (π/4 - 1/2 ln(2)) = π/2 - ln(2)
∴ The value of the integral is π/2 - ln(2).
The correct answer is Option (1): π/2 - ln(2)
Evaluate using Special Integral Forms Question 2:
Let
Answer (Detailed Solution Below)
Evaluate using Special Integral Forms Question 2 Detailed Solution
Calculation:
Given,
Let I =
Put x10 = t ⇒ x = t1/10
⇒
∴ I =
⇒ I =
=
=
⇒ a =
⇒ 100(a + b + c) = 100(
∴ The value of 100(a + b + c) is 2120.
The correct answer is Option 4.
Evaluate using Special Integral Forms Question 3:
What is the value of
Answer (Detailed Solution Below)
Evaluate using Special Integral Forms Question 3 Detailed Solution
Concept
Calculation:
Let,
Let f(x) =
⇒
∴
= ex f(x) + c
=
Hence, option (3) is correct.