Question
Integrate the function in Exercise:
$\frac{1}{\text{x}-\text{x}^{3}}$
$\frac{1}{\text{x}-\text{x}^{3}}$
$\Rightarrow1=\text{A}-\text{A}\text{x}^{2}+\text{B}\text{x}+\text{B}\text{x}^{2}+\text{C}\text{x}-\text{C}\text{x}^{2}$
Equating the coefflclents of $\text{x}^{2},\text{x}$ and constant term, we obtain $-\text{A}+\text{B}-\text{C}=0$ $\text{B}+\text{C}=0$ $\text{A}=1$ on solving these equations, we obtain$\text{A}=1,\text{B}=\frac{1}{2},\text{and}\ \text{C}=\frac{1}{2}$
from eqution (1), we obtain $\frac{1}{\text{x}(1-\text{x})(1+\text{x})}=\frac{1}{\text{x}}+\frac{1}{2(1-\text{x})}-\frac{1}{2(1+\text{x})}$ $\Rightarrow\int\frac{1}{\text{x}(1-\text{x})(1+\text{x}}\text{dx}=\int\frac{1}{\text{x}}\text{dx}+\frac{1}{2}\int\frac{1}{1-\text{x}}\text{dx}-\frac{1}{2}\int\frac{1}{1+\text{x}}\text{dx}$ $=\log|\text{x}|-\frac{1}{2}\log|(1-\text{x})|-\frac{1}{2}\log|(1+\text{x)}|$ $=\log|\text{x}|-\log\bigg|(1-\text{x})^{\frac{1}{2}}\bigg|-\log|(1+\text{x)}^{\frac{1}{2}}\bigg|$ $=\log\left|\frac{\text{x}}{(1-\text{x)}^{\frac{1}{2}}(1+\text{x)}^{\frac{1}{2}}}\right|+\text{C}$ $=\log\left|\bigg(\frac{\text{x}^{2}}{1-\text{x}^{2}}\bigg)^{\frac{1}{2}}\right|+\text{C}$ $=\frac{1}{2}\log\left|\frac{\text{x}^{2}}{1-\text{x}}\right|+\text{C}$Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\vec{\text{r}}=\big(4\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}\big)$ and $\vec{\text{r}}=\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}-\mu\big(2\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}\big)$
$\int\frac{\text{x}^2}{\text{x}^2+6\text{x}+12}\text{ dx}$