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in Definite Integrals by (30.0k points)
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Evaluate the following Integral:

\(\int\limits_{0}^{a}\text x\sqrt{\cfrac{a^2-\text x^2}{a^2+\text x^2}}d\text x\)

1 Answer

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Let I =\(\int\limits_{0}^{a}\text x\sqrt{\cfrac{a^2-\text x^2}{a^2+\text x^2}}d\text x\)

As we have the trigonometric identity

to evaluate this integral we use x2 = a2cos 2θ

⇒ 2xdx = –2a2sin(2θ)dθ (Differentiating both sides)

⇒ xdx = –a2sin(2θ)dθ

When x = 0, a2cos 2θ = 0

⇒ cos 2θ = 0 

⇒ 2θ = \(\cfrac{\pi}2\) ⇒ θ = \(\cfrac{\pi}4\) 

When x = a, a2cos 2θ = a2

⇒ cos 2θ = 1

⇒ 2θ = 0 ⇒ θ = 0

So, the new limits are \(\cfrac{\pi}4\) and 0.

Also,

Substituting this in the original integral,

But, we have 2 sin2θ = 1 – cos 2θ

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