Formulae with proofs
Q1.
Sol.
Q2.
Sol.
Q3.
Sol.
Q4.
for x ∈ (– a, a)
Sol.
Q5.
on I,
where I = (a, ∞) or (– ∞, – a)
Sol.
Q6.
on (–a, a)
Sol.
Put x = sin θ for θ ∈
⇒ dx = a cos θ dθ
Hence
Q7.
on [a, ∞)
Sol.
Put x = a cosh θ
⇒ θ = cosh
– 1
⇒ dx = a sinh θ dθ
Q8.
Sol.
Put x = a sinh θ
⇒ θ = sinh
– 1
⇒ dx = a cosh θ dθ
Note:
sinh
–1
+ c
=
+ c
cosh
–1
+ c
=
+ c
sin
–1
+ c
=
– cos
–1
+ c
cos
–1
+ c
=
– sin
–1
+ c
tan
–1
+ c
=
– cot
–1
+ c
cot
–1
+ c
=
– tan
–1
+ c