Question

Evaluate the integral ∫¹ₒ [x²⁄(√4-x²)ᶟ]dx

Answer 1

To solve this integration, we use the method of integration " Trigonometric Substitution "

Let $x=2sin \theta$ , then $dx=2cos \theta d\theta$

Note that when $x=0$ then $\theta =0$

and when $x=1$ , then $\theta=\pi/6$

Let $I=\int\limits_{0}^{1} \frac { x^{2}} {(\sqrt4-x^{2} )^{3}} dx$ , then $I=\int\limits_{0}^{\pi/6} \frac { 4sin^{2}x} {8cos^{3}\theta} *2cos \theta d\theta =\int\limits_{0}^{\pi/6} tan^{2}\theta d\theta$

Thus $I=\int\limits_{0}^{\pi/6} (sec^{2}\theta-1)d\theta=[tan\theta-\theta]^{\pi/6}_{0}$

and then $I=[tan\theta-\theta]^{\pi/6}_{0}=(1/\sqrt3)-\pi/6$

Therefore we have that

$\int\limits_{0}^{1} \frac { x^{2}} {(\sqrt4-x^{2} )^{3}} dx=(1/\sqrt3)-(\pi/6)$