$\int \frac{cos 2 x + x + 1}{x^{2} + s i n 2 x + 2 x} d x$

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Solution

Consider the given integral.

$\int \frac{c o s 2 x + x + 1}{x^{2} + s i n 2 x + 2 x} d x$

$t = x^{2} + s i n 2 x + 2 x$

Differentiate w.r.t $x$ , we get.

$\frac{d t}{d x} = 2 x + 2 cos 2 x + 2$

$\frac{d t}{2 x + 2 cos 2 x + 2} = d x$

$\frac{d t}{2 (x + cos 2 x + 1)} = d x$

Integrate w.r.t $t$ , we get

$= \frac{1}{2} \int \frac{(c o s 2 x + x + 1 x)}{t} \times \frac{d t}{(x + cos 2 x + 1)}$

$1 = \frac{1}{2} \int \frac{1}{t} d t$

$= \frac{1}{2} ln t + c$

$= \frac{1}{2} ln (x + cos 2 x + 1) + c$

Hence, this is the answer.

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