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RS Aggarwal Class 12 Chapter 14 Solutions (Some Special Integrals)

RS Aggarwal Solutions for Class 12 Chapter 14 ‘Some Special Integrals’ are prepared with the viewpoint of Class 12 CBSE board exam and other competitive entrance exams such as JEE or NEET. Therefore, these solutions are very much compatible with your level of understanding since they are given in a step by step manner with proper reasoning in each step. You will learn the standard 6 integration formulae of some particular functions along with their proofs. You will also be able to derive further identities using these formulae.

There are a sum total of 3 exercises consisting of 55 questions covering all the special integration formulae thoroughly from the exam point of view. The questions range from objective type or very short answer type to long answers which will help you score good marks in your school exams and will also help you to get through the entrance exam of your dream college. The questions in RS Aggarwal Solutions for Class 12 Chapter 14 are written keeping in mind the latest exam pattern of the board and entrance exam syllabus.

The maths experts at Instasolv have created a set of answers in the simplest language possible. You will be at ease while doing your homework using the Instasolv platform as your source of reference. Instasolv is committed to helping you use your study time both effectively as well as efficiently. Therefore, it advised that you add Instasolv’s RS Aggarwal Solutions in your daily study schedule.

Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 14 Some Special Integrals

Integral of Some Particular Functions:

There are a few key integration formulas that are used for integrating various other standard integrals. Here, we will take a look at these, prove them and use them.

Integral of Some Particular Functions

Look at the following integration formulas

  1. ∫ dx / (x2 – a2) = 1/2a log |(x – a) / (x + a)| + C
  2. ∫ dx / (a2 – x2) = 1/2a log |(a + x) / (a – x)| + C
  3. ∫ dx / (x2 + a2) = 1/a tan–1 (x/a) + C
  4. ∫ dx / √ (x2 – a2) = log |x + √ (x2 – a2)| + C
  5. ∫ dx / √ (a2 – x2) = sin–1 (x/a) + C
  6. ∫ dx / √ (x2 + a2) = log |x + √ (x2 + a2)| + C

Proof of the Above Six Standard Integration Formulas:

  •  ∫ dx / (x2 – a2) = ½ a log |(x – a) / (x + a)| + C

We know that,

1 / (x2 – a2) = 1 / (x – a) (x + a) = 1/2a [(x + a) – (x – a) / (x – a) (x + a)]

= 1/2a [1/(x – a) – 1/(x + a)]

Therefore,

∫ dx / (x2 – a2) = ½ a [∫ dx / (x – a) – ∫ dx / (x + a)]

= 1/2a [log |(x – a) – log |(x + a)] + C

= 1/2a log |(x – a) / (x + a)| + C

  • ∫ dx / (a2 – x2) = ½ a log |(a + x) / (a – x)| + C

We know that,

1 / (a2 – x2) = 1 / (a – x) (a + x) = ½ a [(a + x) + (a – x) / (a – x) (a + x)]

= ½ a [1/(a – x) + 1/(a + x)]

Therefore,

∫ dx / (a2 – x2) = ½ a [∫ dx / (a – x) + ∫ dx / (a + x)]

= ½ a [– log |(a – x) + log |(a + x)] + C

= ½ a log |(a + x) / (a – x)| + C

  • ∫ dx / (x2 + a2) = 1/a tan–1 (x/a) + C

substituting x = a tan t, so we have dx = a sec2 t dt. Therefore,

∫ dx / (x2 + a2) = ∫ [(a sec2 t dt) / (a2 tan2 t + a2)]

Solving this, we get,

∫ dx / (x2 + a2) = 1/a ∫ dt = t/a + C

Re-substituting the value of t, we get

∫ dx / (x2 + a2) = 1/a tan–1 (x/a) + C

Similarly, We can also derive other standard formulae.

Highlights of Some Special Integrals:

We can use the above standard formulae to find the following:

  • ∫ dx / (ax. + bx + c)
  • ∫ [(px + q) / (ax2 + bx + c)] dx 

where p, q, a, b, and c are constants.

Exercise-Wise Discussion of RS Aggarwal Solutions for Class 12 Chapter 14-Some Special Integrals

  1. All the exercises in this chapter comprise thorough miscellaneous questions related to the standard formulae given in this chapter.
  2. In these exercise solutions, you will either be required to evaluate some complex integral functions or you will be needed to derive some important results which you may employ in the JEE exam questions.
  3. These questions are of a wide variety from beginners’ level to advanced level questions arranged in an order of increasing difficulty level.
  4. The exercise solutions are covered in a detailed way mentioning all the reasons used in the algorithm, hence, enhancing your timings to approach the questions in maths.
  5. The solutions are mentioned in simple language by the experts at Instasolv.
  6. We have covered each and every question without compromising with the quality of answers.

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