Evaluate the following integrals

Let

Let x=a tan²y

⇒ dx=2a tan y sec²y dy.

Also, when x=0, y=0

and when x=a,

Hence

Using integration by parts, we get

Let tan y=t

⇒ sec²ydy=dt.

Also, when y=0, t=0

and when , t=1.

Also, y=tan^-1t

Let

Hence

Substituting value of I’ in I, we get