الكسور الجزئية - رياضيات متقدم عمر منصور

1. 1)  $\int \frac{2x^2}{x^2-1}dx$

   • $\ln \left|\frac{x-1}{x+1}\right|+c$

   • $2x+\ln \left|\frac{x-1}{x+1}\right|+c$

   • $x+\ln \left|\frac{x-1}{x+1}\right|+c$

   • $2x-\ln \left|\frac{x-1}{x+1}\right|+c$

2. 2)  ${\int }_0^2\frac{x-10}{(x-1){(x+2)}^2}dx$

   • $1-\ln 2$

   • $1+\ln 2$

   • $\ln 2$

   • $\ln 4$

3. 3)  $\int \frac{2}{x(x^2+1)}dx$

   • $2 \ln \left|x\right|+\ln |x^2+1|+c$

   • $\ln \left|x\right|+\ln |x^2+1|+c$

   • $-\ln |x^3+x|+c$

   • $2 \ln \left|x\right|-\ln |x^2+1|+c$

4. 4) إذا كان $\int \frac{8}{u^2-4}du=b$ ، فإن $\int \frac{\sqrt{x}}{x-4}dx$ بدلالة $b$

   • $2b$

   • $2x+b$

   • $2\sqrt{x}+b$

   • $b$

5. 5) اذا كان $b={\int }_2^5\frac{1}{(x-1){(x+3)}^2}$ فإن ${\int }_2^5\frac{x}{(x-1){(x+3)}^2}dx$ بدلالة $b$

   • $\frac{3}{40}+b$

   • $1+b$

   • $2b$

   • $b$

6. 6)  $\int \frac{2 se{c}^{2}x}{se{c}^{2}x-2}dx$

   • $\ln \left|\tan \right(x)-1|-\ln \left|\tan \right(x)+1|+c$

   • $\ln \left|\tan \right(x)+1|+\ln \left|\tan \right(x)-1|+c$

   • $\ln \left|sec\right(x)-1|-\ln \left|sec\right(x)+1|+c$

   • $\ln \left|\tan \right(x\left)\right|-\ln |1+\tan (x\left)\right|+c$

7. 7)  $\int \sqrt{4-e^x}dx$

   • $2\sqrt{4-e^x}-2 \ln \left|\frac{\sqrt{4-e^x}-2}{\sqrt{4-e^x}+2}\right|+c$

   • $\sqrt{4-e^x}+2 \ln \left|\frac{\sqrt{4-e^x}+2}{\sqrt{4-e^x}-2}\right|+c$

   • $2\sqrt{4-e^x}+2 \ln \left|\frac{\sqrt{4-e^x}-2}{\sqrt{4-e^x}+2}\right|+c$

   • $2 \ln \left|\frac{\sqrt{4-e^x}-2}{\sqrt{4-e^x}+2}\right|+c$

8. 8)  ${\int }_{\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{6}}\frac{-2 cot(-x)}{(\ln {(\sin x))}^2-1}dx$

   • $\ln \left(\frac{1-\ln 2}{1+\ln 2}\right)$

   • $\ln \left(\frac{1+\ln 2}{1-\ln 2}\right)$

   • $\ln \left(\frac{\ln \left(2\right)-1}{\ln 2+1}\right)$

   • $\ln (1+\ln 2)$

9. 9)  ${\int }_0^{\ln 3}\frac{4e^x}{e^{2x}+4}dx$

   • $\ln (0.6)$

   • $\ln \frac{5}{3}$

   • $\ln 15$

   • $\ln 2$

10. 10)  $\int \frac{x^2-x+1}{x^4+x^2}dx$

    • $\frac{1}{2}\ln \left|\frac{x^2+1}{x^2}\right|+ \frac{1}{x}$

    • $\frac{1}{2}\ln \left|\frac{x^2+1}{x^2}\right|- \frac{1}{x}$

    • $\ln \left|\frac{x^2+1}{x^2}\right|- \frac{1}{x}$

    • $\ln \left|\frac{x^2+1}{x^2}\right|+ \frac{1}{x}$