Given Definite integral can be written as:

(1)

Let us assume y = a²+x²

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

⇒ d(y) = d(a²+x²)

⇒ dy = 2xdx

……(2)

Upper limit for the Definite Integral:

⇒ x=a ⇒ y = a²+a²

⇒ y=2a²……(3)

Lower limit for the Definite Integral:

⇒ x=0 ⇒ y = a²+0²

⇒ y = a²……(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) = (2a² )^1/2 – (a² )^1/2

⇒ I(x) = √2 a – a

⇒ I(x) = a(√2–1)