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40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these?
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40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x...
Integration of (cos x - sin x)/(1 + sin 2x)
To integrate the given expression, we can simplify it by using trigonometric identities and then proceed with the integration.
- We can rewrite sin 2x as 2sin x cos x.
- Therefore, the expression becomes (cos x - sin x)/(1 + 2sin x cos x).
Using Trigonometric Identity
- We can rewrite the denominator as 1 + sin x cos x + sin x cos x.
- This gives us (cos x - sin x)/(1 + sin x cos x + sin x cos x).
- Now, we can rewrite the numerator as (cos x - sin x) = -1(sin x - cos x).
Integration Steps
- Now, our expression becomes -1(sin x - cos x)/(1 + sin x cos x + sin x cos x).
- We can split the denominator into two parts: (1 + sin x cos x) + sin x cos x.
- This allows us to rewrite the expression as -1(sin x - cos x)/[(1 + sin x cos x) + sin x cos x].
- We can further simplify this expression to -1(sin x - cos x)/(1 + sin x cos x) - cos x.
- Now, we can integrate the expression using the substitution method or by recognizing the standard integral forms.
Final Integration
- Integrating the expression, we get (-1/(cos x + sin x)) - arctan(sin x + cos x) + C, where C is the constant of integration.
Therefore, the correct answer is (a) -1/(cos x + sin x) + C.
To integrate the given expression, we can simplify it by using trigonometric identities and then proceed with the integration.
- We can rewrite sin 2x as 2sin x cos x.
- Therefore, the expression becomes (cos x - sin x)/(1 + 2sin x cos x).
Using Trigonometric Identity
- We can rewrite the denominator as 1 + sin x cos x + sin x cos x.
- This gives us (cos x - sin x)/(1 + sin x cos x + sin x cos x).
- Now, we can rewrite the numerator as (cos x - sin x) = -1(sin x - cos x).
Integration Steps
- Now, our expression becomes -1(sin x - cos x)/(1 + sin x cos x + sin x cos x).
- We can split the denominator into two parts: (1 + sin x cos x) + sin x cos x.
- This allows us to rewrite the expression as -1(sin x - cos x)/[(1 + sin x cos x) + sin x cos x].
- We can further simplify this expression to -1(sin x - cos x)/(1 + sin x cos x) - cos x.
- Now, we can integrate the expression using the substitution method or by recognizing the standard integral forms.
Final Integration
- Integrating the expression, we get (-1/(cos x + sin x)) - arctan(sin x + cos x) + C, where C is the constant of integration.
Therefore, the correct answer is (a) -1/(cos x + sin x) + C.
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40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these?
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40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + 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 40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these?.
40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + 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 40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these?.
Solutions for 40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these? in English & in Hindi are available as part of our courses for UPSC.
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40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these?, a detailed solution for 40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these? has been provided alongside types of 40. integrate (cos x - sin x)/(1 + sin 2x) dx = (a) - 1/(cos x + sin x) + c (c) 1/(cos x - sin x) + c (b) 1/(cos x + sin x) + c (d) None of these? theory, EduRev gives you an
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