UPSC Exam > UPSC Questions > 85. integrate arcsin(3x - 4x ^ 3) dx = (a) x ...
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85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c?
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85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ...
Integration of arcsin(3x - 4x^3) dx
To integrate the given function, we need to use integration by parts method. Let's break down the solution step by step:
Step 1: Identify u and dv
- Let u = arcsin(3x - 4x^3)
- dv = dx
Step 2: Find du and v
- du = (1/sqrt(1 - (3x - 4x^3)^2)) * (3 - 12x^2) dx
- v = x
Step 3: Apply integration by parts formula
∫ u dv = uv - ∫ v du
Step 4: Substitute u, v, du, and dv into the formula
- uv = x * arcsin(3x - 4x^3)
- ∫ v du = ∫ x * (1/sqrt(1 - (3x - 4x^3)^2)) * (3 - 12x^2) dx
Step 5: Simplify and integrate
- The integral of ∫ x * (1/sqrt(1 - (3x - 4x^3)^2)) * (3 - 12x^2) dx can be quite complex and may not have a simple closed-form solution.
Therefore, the correct answer is (e) None of the above as the integral of arcsin(3x - 4x^3) dx does not simplify to any of the provided options.
To integrate the given function, we need to use integration by parts method. Let's break down the solution step by step:
Step 1: Identify u and dv
- Let u = arcsin(3x - 4x^3)
- dv = dx
Step 2: Find du and v
- du = (1/sqrt(1 - (3x - 4x^3)^2)) * (3 - 12x^2) dx
- v = x
Step 3: Apply integration by parts formula
∫ u dv = uv - ∫ v du
Step 4: Substitute u, v, du, and dv into the formula
- uv = x * arcsin(3x - 4x^3)
- ∫ v du = ∫ x * (1/sqrt(1 - (3x - 4x^3)^2)) * (3 - 12x^2) dx
Step 5: Simplify and integrate
- The integral of ∫ x * (1/sqrt(1 - (3x - 4x^3)^2)) * (3 - 12x^2) dx can be quite complex and may not have a simple closed-form solution.
Therefore, the correct answer is (e) None of the above as the integral of arcsin(3x - 4x^3) dx does not simplify to any of the provided options.
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85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c?
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85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c?.
85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c?.
Solutions for 85. integrate arcsin(3x - 4x ^ 3) dx = (a) x * arcsin(x) + sqrt(1 - x ^ 2) + c (b) x * arcsin(x) - sqrt(1 - x ^ 2) + c (C) 2[x * arcsin(x) + sqrt(1 - x ^ 2)] + c (d) 3[x * arcsin(x) + sqrt(1 - x ^ 2)] + c? in English & in Hindi are available as part of our courses for UPSC.
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