Cho $\int\limits_{1}^{2}{\frac{\text{d}x}{{{x}^{5}}+{{x}^{3}}}}=a.\ln 5+b.\ln 2+c$ với $a,\,\,b,\,\,c$ là các số hữu tỉ. Giá trị của $a+2b+4c$ bằng

Ta có $I=\int\limits_{1}^{2}{\frac{\text{d}x}{{{x}^{5}}+{{x}^{3}}}}=\int\limits_{1}^{2}{\frac{\text{d}x}{{{x}^{3}}\left( {{x}^{2}}+1 \right)}}=\frac{1}{2}\int\limits_{1}^{2}{\frac{2x\,\text{d}x}{{{x}^{4}}\left( {{x}^{2}}+1 \right)}}=\frac{1}{2}\int\limits_{1}^{2}{\frac{\text{d}\left( {{x}^{2}} \right)}{{{x}^{4}}\left( {{x}^{2}}+1 \right)}}=\frac{1}{2}\int\limits_{1}^{4}{\frac{\text{d}t}{{{t}^{2}}\left( t+1 \right)}}$

Xét $\int\limits_{1}^{4}{\frac{\text{d}t}{{{t}^{2}}\left( t+1 \right)}}=\int\limits_{1}^{4}{\frac{t+1-t}{{{t}^{2}}\left( t+1 \right)}\,\text{d}t}=\int\limits_{1}^{4}{\left( \frac{1}{{{t}^{2}}}-\frac{1}{t\left( t+1 \right)} \right)\,\text{d}t}=\int\limits_{1}^{4}{\frac{\text{d}t}{{{t}^{2}}}}-\int\limits_{1}^{4}{\frac{\text{d}t}{t\left( t+1 \right)}}$

$\begin{align}  & =\left. -\frac{1}{t} \right|_{1}^{4}-\int\limits_{1}^{4}{\left( \frac{1}{t}-\frac{1}{t+1} \right)dt=\frac{3}{4}-\left. \left( \ln t-\ln \left( t+1 \right) \right) \right|_{1}^{4}} \\ & =\frac{3}{4}-\ln 4+\ln 5-\ln 2=\frac{3}{4}-3\ln 2+\ln 5. \\\end{align}$

Khi đó $I=\frac{1}{2}\left( \frac{3}{4}-3\ln 2+\ln 5 \right)=\frac{1}{2}.\ln 5-\frac{3}{2}.\ln 2+\frac{3}{8}.$ Vậy $a+2b+4c=-\,1.$