Q. 55.0( 1 Vote )

# Evaluate the following Integrals:

Answer :

Given Definite integral can be written as:

(1)

Let us assume y = a^{2}+x^{2}

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

⇒ d(y) = d(a^{2}+x^{2})

⇒ dy = 2xdx

……(2)

Upper limit for the Definite Integral:

⇒ x=a ⇒ y = a^{2}+a^{2}

⇒ y=2a^{2}……(3)

Lower limit for the Definite Integral:

⇒ x=0 ⇒ y = a^{2}+0^{2}

⇒ y = a^{2}……(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) = (2a^{2} )^{1/2} – (a^{2} )^{1/2}

⇒ I(x) = √2 a – a

⇒ I(x) = a(√2–1)

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