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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:
Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + C
Reason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + c
- a)Both A and R are true and R is the correct explanation of A
- b)Both A and R are true but R is NOT the correct explanation of A
- c)A is true but R is false
- d)A is false and R is True
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Direction: In the following questions, A statement of Assertion (A) is...
∫ex [f(x) + f′(x)]dx = ∫exf(x)dx + ∫exf′(x)dx
= f(x)ex - ∫f′(x)exdx + ∫f′(x)exdx
= ex f(x) + c
Hence R is true.
∫ex(sin x - cos x)dx = ex (-cos) + c
= -ex cos x + c
Hence A is false.
Most Upvoted Answer
Direction: In the following questions, A statement of Assertion (A) is...
Assertion (A): ∫ex[sin x cos x] dx = ex sin x
Reason (R): ∫ex [f(x) f′(x)]dx = ex f(x)
The given assertion states that ∫ex[sin x cos x] dx is equal to ex sin x.
The reason states a general rule that ∫ex [f(x) f′(x)]dx is equal to ex f(x).
Explanation:
To evaluate the given integral, let's use integration by parts.
Integration by parts formula is given by:
∫u dv = uv - ∫v du
Let's assume u = ex and dv = sin x cos x dx.
Differentiating u with respect to x, we get:
du/dx = ex
Integrating dv, we get:
v = -1/2 sin^2 x
Using the integration by parts formula, we have:
∫ex[sin x cos x] dx = ex (-1/2 sin^2 x) - ∫(-1/2 sin^2 x)(ex) dx
Simplifying this expression, we get:
∫ex[sin x cos x] dx = -1/2 ex sin^2 x + 1/2 ∫sin^2 x ex dx
Now, let's evaluate the remaining integral.
Using the identity sin^2 x = (1 - cos 2x)/2, we have:
∫sin^2 x ex dx = ∫(1 - cos 2x)/2 ex dx
Splitting the integral, we get:
∫sin^2 x ex dx = 1/2 ∫ ex dx - 1/2 ∫cos 2x ex dx
Integrating the first term, we have:
1/2 ∫ ex dx = 1/2 ex
Integrating the second term, we have:
-1/2 ∫cos 2x ex dx = -1/4 ∫d(ex) cos 2x = -1/4 ex cos 2x + 1/4 ∫ex sin 2x dx
Now, let's evaluate the remaining integral.
Using the identity sin 2x = 2 sin x cos x, we have:
1/4 ∫ex sin 2x dx = 1/4 ∫ex (2 sin x cos x) dx
Splitting the integral, we get:
1/4 ∫ex (2 sin x cos x) dx = 1/2 ∫ex sin x cos x dx
Substituting this result back into the previous expression, we have:
-1/4 ex cos 2x + 1/4 ∫ex sin 2x dx = -1/4 ex cos 2x + 1/2 ∫ex sin x cos x dx
Now, let's substitute this back into the original expression:
∫ex[sin x cos x] dx = -1/2 ex sin^2 x + 1/2 ∫sin^2 x ex dx
= -1/2 ex sin^2 x + 1/2 (-1/4 ex cos 2x + 1/2 ∫ex sin x cos x dx)
Simplifying this expression, we get:
Reason (R): ∫ex [f(x) f′(x)]dx = ex f(x)
The given assertion states that ∫ex[sin x cos x] dx is equal to ex sin x.
The reason states a general rule that ∫ex [f(x) f′(x)]dx is equal to ex f(x).
Explanation:
To evaluate the given integral, let's use integration by parts.
Integration by parts formula is given by:
∫u dv = uv - ∫v du
Let's assume u = ex and dv = sin x cos x dx.
Differentiating u with respect to x, we get:
du/dx = ex
Integrating dv, we get:
v = -1/2 sin^2 x
Using the integration by parts formula, we have:
∫ex[sin x cos x] dx = ex (-1/2 sin^2 x) - ∫(-1/2 sin^2 x)(ex) dx
Simplifying this expression, we get:
∫ex[sin x cos x] dx = -1/2 ex sin^2 x + 1/2 ∫sin^2 x ex dx
Now, let's evaluate the remaining integral.
Using the identity sin^2 x = (1 - cos 2x)/2, we have:
∫sin^2 x ex dx = ∫(1 - cos 2x)/2 ex dx
Splitting the integral, we get:
∫sin^2 x ex dx = 1/2 ∫ ex dx - 1/2 ∫cos 2x ex dx
Integrating the first term, we have:
1/2 ∫ ex dx = 1/2 ex
Integrating the second term, we have:
-1/2 ∫cos 2x ex dx = -1/4 ∫d(ex) cos 2x = -1/4 ex cos 2x + 1/4 ∫ex sin 2x dx
Now, let's evaluate the remaining integral.
Using the identity sin 2x = 2 sin x cos x, we have:
1/4 ∫ex sin 2x dx = 1/4 ∫ex (2 sin x cos x) dx
Splitting the integral, we get:
1/4 ∫ex (2 sin x cos x) dx = 1/2 ∫ex sin x cos x dx
Substituting this result back into the previous expression, we have:
-1/4 ex cos 2x + 1/4 ∫ex sin 2x dx = -1/4 ex cos 2x + 1/2 ∫ex sin x cos x dx
Now, let's substitute this back into the original expression:
∫ex[sin x cos x] dx = -1/2 ex sin^2 x + 1/2 ∫sin^2 x ex dx
= -1/2 ex sin^2 x + 1/2 (-1/4 ex cos 2x + 1/2 ∫ex sin x cos x dx)
Simplifying this expression, we get:
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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + CReason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + ca)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'D'. Can you explain this answer?
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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + CReason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + ca)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + CReason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + ca)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + CReason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + ca)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'D'. Can you explain this answer?.
Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + CReason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + ca)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + CReason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + ca)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:Assertion (A): ∫ex[sin x - cos x] dx = ex sin x + CReason (R): ∫ex [f(x) +f′(x)]dx =ex f(x) + ca)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'D'. Can you explain this answer?.
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