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38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these?
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38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(ta...
Explanation:
Given Integral:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4))
Key Steps:
1. Use trigonometric identity tan^2 x + 1 = sec^2 x to simplify the integral.
2. Substitute u = tan x and du = sec^2 x dx to further simplify the integral.
3. Integrate the simplified expression to find the final answer.
Detailed Solution:
1. Rewrite the integral using the trigonometric identity:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4)) = ∫(sec^2 x * dx)/(sqrt(sec^2 x + 4)) = ∫(sec^2 x * dx)/(sqrt(sec^2 x + 4))
2. Substitute u = tan x and du = sec^2 x dx:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4)) = ∫(du)/(sqrt(u^2 + 4))
3. Recognize that the integral is in the form of the standard integral for inverse hyperbolic functions:
∫(du)/(sqrt(u^2 + a^2)) = log[u + sqrt(u^2 + a^2)] + C
4. Substitute back u = tan x:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4)) = log[tan x + sqrt(tan^2 x + 4)] + C
Final Answer:
(a) log[tan x + sqrt(tan^2 x + 4)] + C
Therefore, the correct answer is option (a) log[tan x + sqrt(tan^2 x + 4)] + C.
Given Integral:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4))
Key Steps:
1. Use trigonometric identity tan^2 x + 1 = sec^2 x to simplify the integral.
2. Substitute u = tan x and du = sec^2 x dx to further simplify the integral.
3. Integrate the simplified expression to find the final answer.
Detailed Solution:
1. Rewrite the integral using the trigonometric identity:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4)) = ∫(sec^2 x * dx)/(sqrt(sec^2 x + 4)) = ∫(sec^2 x * dx)/(sqrt(sec^2 x + 4))
2. Substitute u = tan x and du = sec^2 x dx:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4)) = ∫(du)/(sqrt(u^2 + 4))
3. Recognize that the integral is in the form of the standard integral for inverse hyperbolic functions:
∫(du)/(sqrt(u^2 + a^2)) = log[u + sqrt(u^2 + a^2)] + C
4. Substitute back u = tan x:
∫(sec^2 x * dx)/(sqrt(tan^2 x + 4)) = log[tan x + sqrt(tan^2 x + 4)] + C
Final Answer:
(a) log[tan x + sqrt(tan^2 x + 4)] + C
Therefore, the correct answer is option (a) log[tan x + sqrt(tan^2 x + 4)] + C.
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38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these?
Question Description
38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these?.
38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 38. int (sec^2 x * dx)/(sqrt(tan^2 x + 4)) = . (a) log[tan x + sqrt(tan^2 x + 4)] + c (b) 1/2 * (log[tan x + sqrt(tan^2 x + 4)]) + c (c) log[1/2 * tan x + 1/2 * sqrt(tan^2 x + 4)] + c (d) None of these?.
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