]]>
LearnNext
Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back

clear

Integrals of some Particular Functions - I

4,954 Views
Have a doubt? Clear it now.
live_help Have a doubt, Ask our Expert Ask Now
format_list_bulleted Take this Lesson Test Start Test

Integrals of some Particular Functions - I - Lesson Summary

Formulae of integrals:

∫dx/(x2 - a2) = 1/2a . log |(x - a)/(x + a)| + C

∫dx/(a2 - x2) = 1/2a . log |(a - x)/(a + x)| + C

∫dx/(x2 + a2) = 1/a . tan-1 x/a + C

1) ∫1/(x2 - a2) dx = ∫1/((x + a)(x - a)) dx


∫1/((x + a)(x - a)) dx

= 1/2a . ∫ [1/(x-a) - 1/(x+a)] dx

= 1/2a . ∫ [1/(x-a)  dx- 1/(x+a) dx]

= 1/2a . [log |x-a| - log |x+a|] + C

= 1/2a . log|(x-a)/(x+a)| + C  ['.' log|a| - log|b| = log|a/b|]


2) ∫ 1/(a2 - x2) dx

∫ 1/((a+x)(a-x)) dx

= 1/2a . ∫[1/(a+x) + 1/(a-x)] dx

1/2a [∫ 1/a+x dx + ∫ 1/a-x dx]

= 1/2a [log|a+x| - log|a-x|] + C

= 1/2a log|(a+x)/(a-x)| + C          ['.' log|a| - log|b| = log|a/b|]


3) dx/(x2 + a2)

Let x = a tan θ

On differentiating both sides, we get

dx = a sec2 θ dθ

∴ ∫ dx/(x2+a2) = ∫ a sec2 θ / (a2 tan2θ + a2)

= ∫ a sec2θ dθ / a2 (tan2θ + 1)

= ∫ a sec2θ dθ / a2 sec2θ            ['.' sec2θ - tan2θ = 1]

= 1/a ∫ dθ

= 1/a . θ + C

= 1/a tan-1 x/a    [x = a tan θ ⇒ θ = tan-1 x/a]


∫dx/(x2 + a2) = 1/a . tan-1 x/a

Comments(0)

Feel the LearnNext Experience on App

Download app, watch sample animated video lessons and get a free trial.

Desktop Download Now
Tablet
Mobile
Try LearnNext at home

Get a free home demo. Book an appointment now!

GET DEMO AT HOME