Evaluate the following Integrals:

Given Definite Integral can be written as:

We know that:

We know that: 1+tan²x = sec²x

Let us assume,

Differentiating w.r.t x on both sides we get,

Upper limit for the Definite Integral:

⇒ y = 1+∞

⇒ y = ∞……(3)

Lower limit for the Definite Integral:

⇒ x = 0 ⇒ y = 1+tan(0)

⇒ y = 1+0

⇒ y = 1……(4)

Substituting (2),(3),(4) in the eq(1), we get,

We know that:

We know that:

[here f’(x) is derivative of f(x)).

⇒ I(x) = tan^-1(∞) – tan^-1(0)