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Question Number 44306 by abdo.msup.com last updated on 26/Sep/18

$$find\int \frac{dt}{(t+1)\sqrt{t}+t\sqrt{t+1}}$$$$2)calculate{\int }_1^3\frac{dt}{(t+1)\sqrt{t}+t\sqrt{t+1}}$$

Commented by maxmathsup by imad last updated on 29/Sep/18

$$I=\int \frac{dt}{\sqrt{t}\sqrt{t+1}(\sqrt{t+1}+\sqrt{t})}=\int \frac{\sqrt{t+1}-\sqrt{t}}{\sqrt{t}\sqrt{t+1}}dt$$$$=\int \frac{dt}{\sqrt{t}}-\int \frac{dt}{\sqrt{t+1}}=2\sqrt{t}-2\sqrt{t+1}+c.$$

Commented by maxmathsup by imad last updated on 29/Sep/18

$${\int }_1^3\frac{dt}{(t+1)\sqrt{t}+t\sqrt{t+1}}={[2\sqrt{t}-2\sqrt{t+1}]}_1^3=2\sqrt{3}-4-2+2\sqrt{2}=2\sqrt{2}+2\sqrt{3}-6.$$

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