Question
Evaluate the following integrals:
$\int \frac{\text{x}+1}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
$\int \frac{\text{x}+1}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
✓
Answer
Let $\text{I}=\int \frac{\text{x}+1}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
$\text{I}=\int\frac{(\text{x}-1)+2}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
$\text{I}=\int\frac{\text{dx}}{\sqrt{\text{x}+2}}+2\int\frac{\text{dx}}{(\text{x}+1)\sqrt{\text{x}+2}}\ ...(\text{i})$
Now, $\int\frac{\text{dx}}{\sqrt{\text{x}+2}}+2\sqrt{\text{x}+2}+\text{C}_1$
and, $\int \frac{1}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
Let $\text{x}+2=\text{t}^2$
$\text{dx}=2\text{t dt}$
$\therefore\ \int \frac{1}{(\text{x}-1)\sqrt{\text{x}+2}}=2\int\frac{\text{t dt}}{(\text{t}^2-3)\text{t}}=2\int\frac{\text{dt}}{\text{t}^2-3}$
$=\frac{2\times1}{2\sqrt{3}}\log\bigg|\frac{\text{t}-\sqrt{3}}{\text{t}+\sqrt{3}}\bigg|+\text{C}_2$
$=\frac{1}{\sqrt{3}}\log\bigg|\frac{\sqrt{\text{x}+2}-\sqrt{3}}{\sqrt{\text{x}+2}+\sqrt{3}}\bigg|+\text{C}_2$
Thus, from (i)
$\text{I}=2\sqrt{\text{x}+2}+\text{C}_1+\frac{2}{\sqrt{3}}\log\bigg|\frac{\sqrt{\text{x}+2}-\sqrt{3}}{\sqrt{\text{x}+2}+\sqrt{3}}\bigg|+\text{C}_2$
Hence, $\text{I}=2\sqrt{\text{x}+2}+\frac{2}{\sqrt{3}}\log\bigg|\frac{\sqrt{\text{x}+2}-\sqrt{3}}{\sqrt{\text{x}+2}+\sqrt{3}}\bigg|+\text{C}$
$\text{I}=\int\frac{(\text{x}-1)+2}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
$\text{I}=\int\frac{\text{dx}}{\sqrt{\text{x}+2}}+2\int\frac{\text{dx}}{(\text{x}+1)\sqrt{\text{x}+2}}\ ...(\text{i})$
Now, $\int\frac{\text{dx}}{\sqrt{\text{x}+2}}+2\sqrt{\text{x}+2}+\text{C}_1$
and, $\int \frac{1}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
Let $\text{x}+2=\text{t}^2$
$\text{dx}=2\text{t dt}$
$\therefore\ \int \frac{1}{(\text{x}-1)\sqrt{\text{x}+2}}=2\int\frac{\text{t dt}}{(\text{t}^2-3)\text{t}}=2\int\frac{\text{dt}}{\text{t}^2-3}$
$=\frac{2\times1}{2\sqrt{3}}\log\bigg|\frac{\text{t}-\sqrt{3}}{\text{t}+\sqrt{3}}\bigg|+\text{C}_2$
$=\frac{1}{\sqrt{3}}\log\bigg|\frac{\sqrt{\text{x}+2}-\sqrt{3}}{\sqrt{\text{x}+2}+\sqrt{3}}\bigg|+\text{C}_2$
Thus, from (i)
$\text{I}=2\sqrt{\text{x}+2}+\text{C}_1+\frac{2}{\sqrt{3}}\log\bigg|\frac{\sqrt{\text{x}+2}-\sqrt{3}}{\sqrt{\text{x}+2}+\sqrt{3}}\bigg|+\text{C}_2$
Hence, $\text{I}=2\sqrt{\text{x}+2}+\frac{2}{\sqrt{3}}\log\bigg|\frac{\sqrt{\text{x}+2}-\sqrt{3}}{\sqrt{\text{x}+2}+\sqrt{3}}\bigg|+\text{C}$
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