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$\int\limits_{1}^{3}$$\frac{dx}{\sqrt{( 1 + x )^{3}\times ( 3x + 1 )}}$ = $\int\limits_{1}^{3}$$ \frac{ dx }{\sqrt{(( 1 + x )^{4}( \frac{ 3x + 1}{ 1 + x})}}$ = $\int\limits_{1}^{3}$$\frac{dx}{( 1 + x )^{2}\sqrt{\frac{ 3x + 1}{ 1 + x}}}$ (1)
Đặt t = $\sqrt{\frac{ 3x + 1}{ 1 + x}}$ $\Rightarrow $ $t^{2}$ = $\frac{ 3x + 1}{ 1 + x}$ $\Rightarrow $ 2tdt = $\frac{2}{( 1 + x )^{2}}$ dx
Đổi cận : x = 1 $\rightarrow $ t =$ \sqrt{2}$
x = 3 $\rightarrow $ t = $\sqrt{\frac{5}{2}}$
(1) $\Leftrightarrow $ $\int\limits_{\sqrt{2}}^{\sqrt{\frac{5}{2}}}$$\frac{t}{t}$ dt $\Rightarrow $ $\int\limits_{1}^{3}$$\frac{dx}{\sqrt{(( 1 + x)^{3}( 3x + 1)}}$ = $\sqrt{\frac{5}{2}}$ - $\sqrt{2}$
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