Trigonometry Everywhere - 2

\[I=\int\dfrac{2xdx}{(x^2+1)(x^2+3)}\] \[let\space x=\red{\tan\theta}\Rightarrow dx=\sec^2\theta d\theta\] \[\Rightarrow I=2\int\dfrac{\tan\theta\sec^2\theta d\theta}{(\tan^2\theta+1)(\tan^2\theta+3)}\] \[=2\int\dfrac{\tan\theta\cancel{\sec^2\theta}d\theta}{\cancel{\sec^2\theta}(\sec^2\theta+2)}\] \[=2\int\dfrac{\tan\theta\red{\times\cos^2\theta} d\theta}{(\sec^2\theta+2)\red{\times\cos^2\theta}}\] \[=2\int\dfrac{\sin\theta\cos\theta d\theta}{1+2\cos^2\theta}\] \[=\int\dfrac{\sin 2\red{\theta} d\red{\theta}}{2+\cos 2\red{\theta}}\] \[=\dfrac{1}{2}\int\dfrac{\sin \red{u} d\red{u}}{2+\cos \red{u}}\] \[=\dfrac{1}{2}\int\dfrac{\red{\sin {u} d{u}}}{2+\cos {u}}\] \[=\dfrac{-1}{2}\int\dfrac{{d\red{\cos u}}}{2+\red{\cos {u}}}\] \[=\dfrac{-1}{2}\ln(| 2+\red{\cos u}|+C\] \[=\dfrac{-1}{2}\ln(|1+\red{1+\cos 2\theta}|)+C\] \[=\dfrac{-1}{2}\ln(|1+\red{2\cos^2 \theta}|)+C\] \[=\dfrac{-1}{2}\ln(|1+{\dfrac{2}{1+\red{\tan^2\theta}}}|)+C\] \[=\dfrac{-1}{2}\ln(|1+{\dfrac{2}{1+\red{x^2}}}|)+C\]

#Calculus

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$