Evaluate the following Integrals:

Let

Put x = atan²θ

⇒ x = 2a tan θ sec²θ dθ (Differentiating both sides)

When x = 0, atan²θ = 0 ⇒ tan θ = 0 ⇒ θ = 0

So, the new limits are 0 and.

Also,

We have the trigonometric identity 1 + tan²θ = sec²θ

Substituting this in the original integral,

Now, put tan θ = t

⇒ sec²θ dθ = dt (Differentiating both sides)

When θ = 0, t = tan 0 = 0

So, the new limits are 0 and 1.

Substituting this in the original integral,

We will use integration by parts.

Recall

Here, take f(t) = tan^-1t and g(t) = t

Now,

Substituting these values, we evaluate the integral.

We can write

Recall