Question

integration of {√(1+√x)}dx from (0 to 1)

Answer 1

$Let\;\;I=\int\limits^1_0{\sqrt{1+\sqrt{x}}} \, dx\\\;\\let\;x=t^{2}\to\;dx=2tdt\\\;\\when,x=0,t=0\;and\; x=1,t=1\\\;\\I=\int\limits^1_0 {2t.\sqrt{1+t^{2}} }} \, dt\\\;\\let\; t=\tan\theta\to\;dt=\sec^{2}\theta\;d\theta\\\;\\t=0,\theta=0\;and\;t=1,\theta=\frac{\pi}{4}\\\;\\I=2\int\limits^\frac{\pi}{4}_0 {\sec^{2} \theta.\tan\theta\;sec\theta} \, d\theta\\\;\\I=\frac{2}{3}[\sec^{3}\theta]\limits^\frac{\pi}{4}_0} \\\;\\I=\frac{2}{3}(\sec^{3}\frac{\pi}{4}-1)\\\;\\I=\frac{2}{3}(2\sqrt{2}-1)$