tính tích phân

Question

$\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}}\frac{dx}{\\sin ^{2}x +3\sin x\cos x+2\cos ^{2}x}$

score 1 · Answer 1

$I=\int\limits_{}^{}\frac{dx}{sin^2x+3sinxcosx+2cos^2x}=\int\limits_{}^{}\frac{dx}{cos^2(tan^2x+3tanx+2)}$
$=\int\limits_{}^{}\frac{d(tanx)}{tan^2x+3tanx+2}=\int\limits_{}^{}\frac{dt}{t^2+3t+2}$
$=\int\limits_{}^{}(\frac{dt}{t+1}-\frac{dt}{t+2})=ln\left| {\frac{t+1}{t+2}} \right|$
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