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Question Number 82577 by M±th+et£s last updated on 22/Feb/20

$$\int \frac{2-\sqrt{x+3}}{2+\sqrt{x-3}}dx$$

Commented by mathmax by abdo last updated on 22/Feb/20

$$letI=\int \frac{2-\sqrt{x+3}}{2+\sqrt{x+3}}dxvhangement2+\sqrt{x+3}=tgive$$$$\sqrt{x+3}=t-2\Rightarrow x+3={(t-2)}^2=t^2-4t+4\Rightarrow x=t^2-4t+1\Rightarrow$$$$I=\int \frac{2-\sqrt{t^2-4t+1+3}}{t}(2t-4)dt=2\int \frac{(t-2)\sqrt{t^2-4t+4}}{t}dt$$$$=2\int \frac{(t-2)(t-2)}{t}dt=2\int \frac{t^2-4t+4}{t}dt=2\int (t-4+\frac{4}{t})dt$$$$=t^2-8t+8ln\mid t\mid +C$$$$={(2+\sqrt{x+3})}^2-8(2+\sqrt{x+3})+8ln\mid 2+\sqrt{x+3}\mid +C$$

Commented by M±th+et£s last updated on 22/Feb/20

$$godblessyousir$$

Commented by msup trace by abdo last updated on 22/Feb/20

$$thankyousir.$$

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