$\int \frac{\cos 2 x + x + 1}{x^{2} + \sin 2 x + 2 x} d x$

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Solution

$Let I = \int \frac{\cos 2 x + x + 1}{x^{2} + \sin 2 x + 2 x} d x Putting x^{2} + \sin 2 x + 2 x = t \Rightarrow 2 x + 2 \cos 2 x + 2 = \frac{d t}{d x} \Rightarrow (x + \cos 2 x + 1) d x = \frac{d t}{2} ∴ I = \frac{1}{2} \int \frac{1}{t} d t = \frac{1}{2} \ln |t| + C = \frac{1}{2} \ln |x^{2} + \sin 2 x + 2 x| + C [∵ t = x^{2} + \sin 2 x + 2 x]$

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