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Test: Integrals- Case Based Type Questions - JEE MCQ


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15 Questions MCQ Test Mathematics (Maths) Class 12 - Test: Integrals- Case Based Type Questions

Test: Integrals- Case Based Type Questions for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The Test: Integrals- Case Based Type Questions questions and answers have been prepared according to the JEE exam syllabus.The Test: Integrals- Case Based Type Questions MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Integrals- Case Based Type Questions below.
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Test: Integrals- Case Based Type Questions - Question 1

Read the following text and answer the following questions on the basis of the same:

∫ ex[f(x) +f′(x)]dx = ∫ exf(x)dx + ∫ex f′(x)dx

= f(x)ex – ∫ f ′(x)ex dx + ∫ f ′(x)ex dx

= ex f(x) + c

Q. ∫ ex (sinx + cos x)dx = _______.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 1

Test: Integrals- Case Based Type Questions - Question 2

Read the following text and answer the following questions on the basis of the same:

∫ ex[f(x) +f′(x)]dx = ∫ exf(x)dx + ∫ex f′(x)dx

= f(x)ex – ∫ f ′(x)ex dx + ∫ f ′(x)ex dx

= ex f(x) + c

Q. ∫ ex (x + 1) dx = _______.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 2

Test: Integrals- Case Based Type Questions - Question 3

Read the following text and answer the following questions on the basis of the same:

∫ ex[f(x) +f′(x)]dx = ∫ exf(x)dx + ∫ex f′(x)dx

= f(x)ex – ∫ f ′(x)ex dx + ∫ f ′(x)ex dx

= ex f(x) + c

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 3

Test: Integrals- Case Based Type Questions - Question 4

Read the following text and answer the following questions on the basis of the same:

∫ ex[f(x) +f′(x)]dx = ∫ exf(x)dx + ∫ex f′(x)dx

= f(x)ex – ∫ f ′(x)ex dx + ∫ f ′(x)ex dx

= ex f(x) + c

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 4

Test: Integrals- Case Based Type Questions - Question 5

Read the following text and answer the following questions on the basis of the same:

∫ ex[f(x) +f′(x)]dx = ∫ exf(x)dx + ∫ex f′(x)dx

= f(x)ex – ∫ f ′(x)ex dx + ∫ f ′(x)ex dx

= ex f(x) + c

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 5

Test: Integrals- Case Based Type Questions - Question 6

Read the following text and answer the following questions on the basis of the same:

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 6
Since, xsin x is an even function

Test: Integrals- Case Based Type Questions - Question 7

Read the following text and answer the following questions on the basis of the same:

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 7

since xcos x is an odd function.

Test: Integrals- Case Based Type Questions - Question 8

Read the following text and answer the following questions on the basis of the same:

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 8

since sin3 x is an odd function.

Test: Integrals- Case Based Type Questions - Question 9

Read the following text and answer the following questions on the basis of the same:

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 9

Since it is an odd function

Test: Integrals- Case Based Type Questions - Question 10

Read the following text and answer the following questions on the basis of the same:

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 10

since x99 is an odd function.

Test: Integrals- Case Based Type Questions - Question 11

Read the following text and answer the following questions on the basis of the same:

Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:

In the above table, A, B and C are real numbers to be determined suitably.

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 11
Let ex = t

⇒ ex dx = dt

1 = A(t - 1) + Bt …..(i)

Substituting t = 1 and t = 0 in equation (i), we obtain

A = –1 and B = 1

Test: Integrals- Case Based Type Questions - Question 12

Read the following text and answer the following questions on the basis of the same:

Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:

In the above table, A, B and C are real numbers to be determined suitably.

Q. Integration of

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 12

Equating the coefficients of x and constant term, we obtain

A + B = 1

2A + B = 0

On solving, we obtain A = -1 and B = 2

= -log|x+1|+2log|x+2|+C

= log(x +2)2 - log|x+1|+C

Test: Integrals- Case Based Type Questions - Question 13

Read the following text and answer the following questions on the basis of the same:

Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:

In the above table, A, B and C are real numbers to be determined suitably.

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 13
We write,

where, real number A and B are to be determined suitably. This gives

1 = A(x+2) + B(x+ 1)

Equating the coefficients of x and the constant term, we get

AB + AB = 0

and 2A + B = 1

Solving these equations, we get A = 1 and B = –1.

Thus, the integrand is given by

Therefore,

Test: Integrals- Case Based Type Questions - Question 14

Read the following text and answer the following questions on the basis of the same:

Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:

In the above table, A, B and C are real numbers to be determined suitably.

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 14

Equating the coefficients of x and constant term, we obtain

A + B = 0

-3A + 3B = 1

On solving, we obtain

Test: Integrals- Case Based Type Questions - Question 15

Read the following text and answer the following questions on the basis of the same:

Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:

In the above table, A, B and C are real numbers to be determined suitably.

Q.

Detailed Solution for Test: Integrals- Case Based Type Questions - Question 15

Equating the coefficients of x2, x and constant term, we obtain

A + B = 0

C = 0

A = 1

On solving these equations, we obtain

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