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