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Question Number 55089 by eso last updated on 17/Feb/19

$$\underset{0}{\stackrel{1}{\int }}\frac{1}{{(x^2+1)}^{3/2}}dx=$$

Commented by maxmathsup by imad last updated on 17/Feb/19

$$letI={\int }_0^1\frac{dx}{{(1+x^2)}^{\frac{3}{2}}}changementx=tantgive$$$$I={\int }_0^{\frac{\pi }{4}}\frac{1+{tan}^{2}t}{{(1+{tan}^{2}t)}^{\frac{3}{2}}}dt={\int }_0^{\frac{\pi }{4}}\frac{dt}{{(1+{tan}^{2}t)}^{\frac{1}{2}}}={\int }_0^{\frac{\pi }{4}}{\left({cos}^{2}t\right)}^{\frac{1}{2}}dt$$$$={\int }_0^{\frac{\pi }{4}}costdt={\left[sint\right]}_0^{\frac{\pi }{4}}=\frac{\sqrt{2}}{2}$$

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