# NCERT Solutions for Exercise 7.5 Class 12 Maths Chapter 7 - Integrals

NCERT solutions for Class 12 Maths chapter 7 exercise 7.5 deals with some of the functions which are not discussed yet in earlier exercises. It includes rational functions and advanced levels of logarithmic functions. If students practice this NCERT book exercise diligently, they can attain a good level of understanding of Integrals. Exercise 7.5 Class 12 Maths questions can be seen verbatim in CBSE board examinations. NCERT solutions for Class 12 Maths chapter 7 exercise 7.5 along with some in text examples is recommended to be solved . You can have a look at the NCERT exercises provided below.

Integrals Exercise 7.1

Integrals Exercise 7.2

Integrals Exercise 7.3

Integrals Exercise 7.4

Integrals Exercise 7.6

Integrals Exercise 7.7

Integrals Exercise 7.8

Integrals Exercise 7.9

Integrals Exercise 7.10

Integrals Exercise 7.11

Integrals Miscellaneous Exercise

**Integrals****Class 12**

**Chapter 7**

**Exercise: 7.5**

**Question:1 ** Integrate the rational functions

** Answer: **

Given function

Partial function of this function:

Now, equating the coefficients of x and constant term, we obtain

On solving, we get

### ** Question:2 ** Integrate the rational functions

### ** Answer: **

Given function

The partial function of this function:

Now, equating the coefficients of x and constant term, we obtain

On solving, we get

### ** Question:3 ** Integrate the rational functions

### ** Answer: **

Given function

Partial function of this function:

** .(1) **

** Now, substituting respectively in equation (1), we get **

** That implies **

### ** Question:4 ** Integrate the rational functions

### ** Answer: **

Given function

Partial function of this function:

** .....(1) **

** Now, substituting respectively in equation (1), we get **

** That implies **

### ** Question:5 ** Integrate the rational functions

### ** Answer: **

Given function

Partial function of this function:

** ...........(1) **

** Now, substituting respectively in equation (1), we get **

** That implies **

### ** Question:6 ** Integrate the rational functions

### ** Answer: **

Given function

Integral is not a proper fraction so,

Therefore, on dividing by , we get

Partial function of this function:

** ...........(1) **

** Now, substituting respectively in equation (1), we get **

No, substituting in equation (1) we get

### ** Question:7 ** Integrate the rational functions

### ** Answer: **

Given function

Partial function of this function:

Now, equating the coefficients of and the constant term, we get

** and **

** On solving these equations, we get **

** From equation (1), we get **

Now, consider ,

and we will assume

So,

or

### ** Question:8 ** Integrate the rational functions

### ** Answer: **

Given function

Partial function of this function:

Now, putting in the above equation, we get

By equating the coefficients of and constant term, we get

then after solving, we get

Therefore,

### ** Question:9 ** Integrate the rational functions

### ** Answer: **

Given function

can be rewritten as

Partial function of this function:

** ................(1) **

Now, putting in the above equation, we get

By equating the coefficients of and , we get

then after solving, we get

Therefore,

### ** Question:10 ** Integrate the rational functions

### ** Answer: **

Given function

can be rewritten as

The partial function of this function:

Equating the coefficients of , we get

Therefore,

### ** Question:11 ** Integrate the rational functions

### ** Answer: **

Given function

can be rewritten as

The partial function of this function:

Now, substituting the value of respectively in the equation above, we get

Therefore,

### ** Question:12 ** Integrate the rational functions

### ** Answer: **

Given function

As the given integral is not a proper fraction.

So, we divide by , we get

can be rewritten as

** ....................(1) **

** Now, substituting in equation (1), we get **

Therefore,

### ** Question:13 ** Integrate the rational functions

### ** Answer: **

Given function

can be rewritten as

** ....................(1) **

Now, equating the coefficient of and constant term, we get

, , and

Solving these equations, we get

Therefore,

### ** Question:14 ** Integrate the rational functions

### ** Answer: **

Given function

can be rewritten as

** **

Now, equating the coefficient of and constant term, we get

and ,

Solving these equations, we get

Therefore,

### ** Question:15 ** Integrate the rational functions

### ** Answer: **

Given function

can be rewritten as

The partial fraction of above equation,

Now, equating the coefficient of and constant term, we get

and

and

Solving these equations, we get

Therefore,

### ** Question:16 ** Integrate the rational functions

[Hint: multiply numerator and denominator by and put ]

### ** Answer: **

Given function

Applying Hint multiplying numerator and denominator by and putting

Putting

can be rewritten as

Partial fraction of above equation,

** ................(1) **

** Now, substituting in equation (1), we get **

### ** Question:17 ** Integrate the rational functions

[Hint : Put ]

### ** Answer: **

Given function

Applying the given hint: putting

We get,

Partial fraction of above equation,

** ................(1) **

** Now, substituting in equation (1), we get **

Back substituting the value of t in the above equation, we get

### ** Question:18 ** Integrate the rational functions

### ** Answer: **

Given function

We can rewrite it as:

Partial fraction of above equation,

** **

** Now, equating the coefficients of and constant term, we get **

** , , , **

** After solving these equations, we get **

### ** Question:19 ** Integrate the rational functions

### ** Answer: **

Given function

Taking

The partial fraction of above equation,

** ..............(1) ** ** **

** Now, substituting in equation (1), we get **

### ** Question:20 ** Integrate the rational functions

### ** Answer: **

Given function

So, we multiply numerator and denominator by , to obtain

Now, putting

we get,

Taking

Partial fraction of above equation,

** ..............(1) ** ** **

** Now, substituting in equation (1), we get **

** Back substituting the value of t, **

### ** Question:21 ** Integrate the rational functions [Hint : Put ]

### ** Answer: **

Given function

So, applying the hint: Putting

Then

Partial fraction of above equation,

** ..............(1) ** ** **

** Now, substituting in equation (1), we get **

** Now, back substituting the value of t, **

### ** Question:22 ** Choose the correct answer

### ** Answer: **

Given integral

Partial fraction of above equation,

** ..............(1) ** ** **

** Now, substituting in equation (1), we get **

** Therefore, the correct answer is B. **

### ** Question:23 ** Choose the correct answer

### ** Answer: **

Given integral

Partial fraction of above equation,

** **

** Now, equating the coefficients of and the constant term, we get **

, ,

We have the values,

** Therefore, the correct answer is A. **

**More About NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.5**

The NCERT Class 12 Maths chapter Integrals covers a total of 12 exercises including one Miscellaneous exercise. Exercise 7.5 Class 12 Maths has a total of 23 main questions along with some few subquestions. In NCERT solutions for Class 12 Maths chapter 7 exercise 7.5 questions difficulty level of questions are of moderate to advanced level which is useful for competitive exams like NEET and JEE Main.

**Also Read| **Integrals Class 12 Notes

**Benefits of NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.5**

The Class 12th Maths chapter 7 exercise is very long. So one should skip some questions to cover maximum syllabus.

Practicing exercise 7.5 Class 12 Maths can certainly help students prepare for Board exams and competitive exams.

These Class 12 Maths chapter 7 exercise 7.5 solutions can be asked directly in the Board exams.

**Also see-**

NCERT Exemplar Solutions Class 12 Maths Chapter 7

NCERT Solutions for Class 12 Maths Chapter 7

**NCERT Solutions Subject Wise**

NCERT Solutions Class 12 Chemistry

NCERT Solutions for Class 12 Physics

NCERT Solutions for Class 12 Biology

NCERT Solutions for Class 12 Mathematics

**Subject Wise NCERT Exemplar Solutions**

NCERT Exemplar Class 12 Maths

NCERT Exemplar Class 12 Physics

NCERT Exemplar Class 12 Chemistry

NCERT Exemplar Class 12 Biology

Happy learning!!!