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[beontop97]

$\int_{0}^{\pi /4} \frac{sinx-\sqrt{2}cosx}{(sinx+cosx)^3}$

[trangxoai1995]

$\int_{0}^{\pi /4} \frac{sinx-\sqrt{2}cosx}{(sinx+cosx)^3}$

$=\int_{0}^{\frac{\pi }{4}}\frac{sinx-\sqrt{2}cosx}{2\sqrt{2}sin^3\left ( x+\frac{\pi }{4} \right )}dx$ (1)

Đặt: $x+\frac{\pi }{4}=t\Rightarrow dx=dt$

$=\int \frac{sin\left ( t-\frac{\pi }{4} \right )-\sqrt{2}cos\left ( t-\frac{\pi }{4} \right )}{2\sqrt{2}sin^3t}dt$

$=\frac{1-\sqrt{2}}{4}\int \frac{1}{sin^2t}dt+\frac{1+\sqrt{2}}{4}\int \frac{d(sint)}{sin^3t}$

Edited by trangxoai1995, 18-05-2014 - 20:56.