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Integrals of Some Particular functions - II

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Integrals of Some Particular functions - II - Lesson Summary

Application in integrating some particular functions.

∫ 1/√(x2 - a2) dx.

∫ 1/√(a2 - x2) dx

∫ 1/√(x2 + a2) dx


1) ∫ 1/√(x2 - a2) dx.

x = a sec θ

dx = a sec θ tan θ dθ

∫ 1/√(x2 - a2) dx = ∫ a sec θ tan θ dθ /√(a2 sec2 θ - a2)

= ∫ a sec θ tan θ dθ /√(a2 sec2 θ - a2)

   ∫ a sec θ tan θ dθ /√(a2 tan2 θ)

   [ using sec2 θ - tan2θ = 1]

= ∫ a sec θ tan θ dθ / a tan θ

= ∫ sec θ dθ

= log|sec θ + tan θ| + P

   x = a sec θ

   sec θ = x/a

= log |x/a + √(x2 - a2)/a| + P

= log |x. √(x2 - a2)/a| + P

= log |x + √(x2 - a2)| - log a + P


   ∫ 1/√(x2 - a2) dx =  log |x + √(x2 - a2)| + C



2) ∫ 1/√(a2 - x2) dx

    x = a sin θ

  dx = a cos θ dθ

∫ 1/√(a2 - x2) dx = ∫ 1/√(a2 - a2 sin2 θ) . a cos θ dθ

= ∫ 1/√(a2 (1 - sin2 θ)) . a cos θ dθ

= ∫ 1/√(a2 cos2 θ) . a cos θ dθ     ['.' sin2 θ + cos2 θ = 1]

= ∫ 1/ (a . cos θ) . a cos θ dθ

= ∫ 1 dθ

= θ + C

   But x = a sin θ

⇒ θ = sin-1 (x/a)

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


3) ∫ 1/√(x2 + a2) dx

   x = a tan θ

 dx = a sec2 θ dθ

= ∫ a sec2 θ dθ/√(a2 tan2θ + a2)

= ∫ a sec2 θ dθ/√(a2 (tan2θ + 1))

= ∫ a sec2 θ dθ/√(a2 sec2θ)

= ∫ a sec2 θ dθ/(a sec θ)

= ∫ sec θ dθ

= log |sec θ + tan θ| + P

= log|√(x2/a2) + 1 + x/a| + P

= log|(√(x2 + a2) + x)/a| + P

= log|x + √(x2 + a2)| - log |a| + P

   ∫ 1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C  Where C = -log|a| + P

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