Question
Evaluate the following intregals: $\int\frac{1}{1-2\sin\text{x}}\ \text{dx}$
✓
Answer
Let $\text{I}=\int\frac{1}{1-2\sin\text{x}}\ \text{dx}$
Put $\sin\text{x}=\frac{2\tan^\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\text{I}=\int\frac{1}{1-2\Bigg(\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)}\ \text{dx}$
$=\int\frac{1+\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}-4\tan\frac{\text{x}}{2}}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{\tan^2\frac{\text{x}}{2}-4\tan\frac{\text{x}}{2}+1}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$=\int\frac{2\text{dt}}{\text{t}^2-4\text{t}+1}$
$=\int\frac{2\text{dt}}{\text{t}^2-2\text{t}(2)+(2)^2-(2)^2+1}$
$\text{I}=2\int\frac{\text{dt}}{(\text{t}-2)^2+(\sqrt{3})^2}$
$=2\times\frac{1}{2\sqrt{3}}\log\Big|\frac{\text{t}-2-\sqrt{3}}{\text{t}-2+\sqrt{3}}\Big|+\text{C}$
$\text{I}=\frac{1}{\sqrt{3}}\log\Bigg|\frac{\tan\frac{\text{x}}{2}-2-\sqrt{3}}{\tan\frac{\text{x}}{2}-2+\sqrt{3}}\Bigg|+\text{C}$
Put $\sin\text{x}=\frac{2\tan^\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\text{I}=\int\frac{1}{1-2\Bigg(\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)}\ \text{dx}$
$=\int\frac{1+\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}-4\tan\frac{\text{x}}{2}}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{\tan^2\frac{\text{x}}{2}-4\tan\frac{\text{x}}{2}+1}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$=\int\frac{2\text{dt}}{\text{t}^2-4\text{t}+1}$
$=\int\frac{2\text{dt}}{\text{t}^2-2\text{t}(2)+(2)^2-(2)^2+1}$
$\text{I}=2\int\frac{\text{dt}}{(\text{t}-2)^2+(\sqrt{3})^2}$
$=2\times\frac{1}{2\sqrt{3}}\log\Big|\frac{\text{t}-2-\sqrt{3}}{\text{t}-2+\sqrt{3}}\Big|+\text{C}$
$\text{I}=\frac{1}{\sqrt{3}}\log\Bigg|\frac{\tan\frac{\text{x}}{2}-2-\sqrt{3}}{\tan\frac{\text{x}}{2}-2+\sqrt{3}}\Bigg|+\text{C}$
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