Integration in spherical coordinates, reduction formula and formula
for integration by substitutions are keys to solve problems in integral
calculus. The formulas are derived stepwise and all relevant information
and skills have been added to make the solution easy and convincing.
Substitute x = a sec 𝜃 in the expression x 2 - a 2
x 2 - a 2 = (a sec 𝜃) 2 - a 2 = a 2 sec 2 𝜃 - a 2 = a 2 sec 2 𝜃 - 1 ------- (2)
Since
1 + tan 2 𝜃 = sec 2 𝜃
tan 2 𝜃 = sec 2 𝜃 - 1 ----------- (3)
Putting equation (3) in equatio (2)
x 2 - a 2 = a 2 tan 2 𝜃
-
x2 - a2 = a 2 tan 2 𝜃 = a tan 𝜃 --------- (4)
Since
x = a sec 𝜃
Differentiating with respect to 𝜃
dx d d
= (a sec 𝜃) = a (sec 𝜃) = a sec 𝜃 tan 𝜃
d𝜃 d𝜃 d𝜃
dx = a sec 𝜃 tan 𝜃 d𝜃 ------------- (5)
Putting equation (4) & equation (5) in equation (1)
a sec 𝜃 tan 𝜃d𝜃
I=∫ = ∫sec 𝜃d𝜃
a tan
I = ln|sec 𝜃 + tan 𝜃| + C1 ---------- (6)
Since
x = a sec 𝜃
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