]> Exercise 3

Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)

Chapter 2: Integration; Section 1: Indefinite Integrals; page 1

Definitions and Basics, Exercise 3

Question

Calculate the following integrals:

  1. ∫ tan⁡ 2 x dx
     
  2. ∫ dx tan⁡ 2 x  
     
  3. ∫ dx 1− x 4  
    Hint for the last case: ( 1− x 4 )=( 1+ x 2 )( 1− x 2 )

Reminder

{ ∫ cos⁡x dx =sin⁡x+C ∫ sin⁡x dx =−cos⁡x+C ∫ ( 1+ tan⁡ 2 x ) dx=∫ dx cos⁡ 2 x =tan⁡x+C ∫ ( 1+ cot⁡ 2 x ) dx=∫ dx sin⁡ 2 x =−cot⁡x+C } (2.1.1.14)
{ ∫ dx 1− x 2 =( or ) { ±asin x+C ∓acos x+C ∫ dx 1+ x 2 =( or ) { +atan x+C −acot x+C } (2.1.1.15)
{ ∫ dx 1+ x 2 =asinh x+C ∫ dx x 2 −1 =∓acosh x+C ∫ dx 1− x 2 =atanh x+C ( | x |<1 ) ∫ dx 1− x 2 =acoth x+C ( | x |>1 ) } (2.1.1.18)

Parts 1-2

Solution of question 1

  1. Among the integrals known so far the ∫ tan⁡ 2 x dx is included in (2.1.1.14): ∫ ( 1+ tan⁡ 2 x )dx =tan⁡x+C
  2. as a consequence: ∫ tan⁡ 2 x  dx=tan⁡x−∫ dx =tan⁡x−x+C

Parts 3-5

Solution of question 2

  1. Among the integrals known so far the ∫ dx tan⁡ 2 x   is not included, but 1 tan⁡x =cot⁡x
     
  2. and therefore (2.1.1.14) could be used: ∫ dx( 1+ cot⁡ 2 x ) =∫ dx( 1+ 1 tan⁡ 2 x ) =− 1 tan⁡x +C
     
  3. As a consequence: ∫ dx tan⁡ 2 x   =− 1 tan⁡x −∫ dx =−( 1 tan⁡x +x )+C

Parts 6-9

Solution of question 3

  1. The integral ∫ dx 1− x 4   does not appear among the known integrals,
    but according to the hint one have to look for the expressions (1+x2) and (1−x2) and luckily they are in (2.1.1.15) and (2.1.1.18).
  2. We have to seek for a possible solution of 1 1− x 4 = A 1+ x 2 + B 1− x 2
  3. Indeed we obtain A 1+ x 2 + B 1− x 2 = ( A+B )+( −A+B ) x 2 1− x 4 and the requirement is fulfilled with A=B= 1 2
  4. Finally we obtain ∫ dx 1− x 4 = 1 2 ∫ dx 1+ x 2 + 1 2 ∫ dx 1− x 2 = = atanx 2 +{ 1 2 atanh x for( | x |<1 ) 1 2 acoth x for( | x |>1 ) }+C

Score

By questions:

Question 1 gives credit of 2 points.
Question 2 gives credit of 3 points.
Question 3 gives credit of 5 points.

By parts:

Parts 1,2,3,4,5,6,8,9 are worth 1 point each.
Part 7 is worth 2 points.