2-1-x-1-x-3-

Question Number 89237 by cindiaulia last updated on 16/Apr/20

\int_{2}^{1} (x + 1) (\sqrt{x + 3)}

Commented by niroj last updated on 16/Apr/20

= \int_{2}^{1} (x + 1) \sqrt{x + 3} dx

put \sqrt{x + 3} = t

x + 3 = t^{2} \Rightarrow x = t^{2} - 3

dx = 2 tdt

if x = 1 \Rightarrow t = 2

if x = 2 \Rightarrow t = \sqrt{5}

\int_{\sqrt{5}}^{2} (t^{2} - 3 + 1) t . 2 tdt

= \int_{\sqrt{5}}^{2} (t^{2} - 2) {2 t}^{2} dt

= \int_{\sqrt{5}}^{2} ({2 t}^{4} - {4 t}^{2}) dt

= {[2 \int t^{4} dt - 4 \int t^{2} dt]}_{\sqrt{5}}^{2}

= {[2 . \frac{t^{5}}{5} - 4 . \frac{t^{3}}{3}]}_{\sqrt{5}}^{2} = {[\frac{2}{5} t^{5} - \frac{4}{3} t^{3}]}_{\sqrt{5}}^{2}

= [\frac{2}{5} {(2)}^{5} - \frac{4}{3} 2^{3}] - [\frac{2}{5} {(\sqrt{5})}^{5} - \frac{4}{3} {(\sqrt{5})}^{3}]

= (\frac{2.32}{5} - \frac{4.8}{3}) - (\frac{2}{5} . 25 \sqrt{5} - \frac{4}{3} . 5 \sqrt{5})

= \frac{64}{5} - \frac{32}{3} - 10 \sqrt{5} + \frac{20 \sqrt{5}}{3}

= \frac{64}{5} - \frac{32}{3} - (\frac{30 \sqrt{5} - 20 \sqrt{5}}{3})

= \frac{64}{5} - \frac{32}{3} - \frac{10 \sqrt{5}}{3} / / .

= 12.8 - 10.67 - 7.453

= 4.677 / / .

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