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 |