Evaluate the following integrals:

Question:

Evaluate the following integrals:

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

Solution:

Assume $x^{2}+\sin 2 x+2 x=t$

$d\left(x^{2}+\sin 2 x+2 x\right)=d t$

$(2 x+2 \cos 2 x+2) d x=d t$

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

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

Put $t$ and $d t$ in given equation we get

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

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

But $t=x^{2}+\sin 2 x+2 x$

$=\frac{1}{2} \ln \left|x^{2}+\sin 2 x+2 x\right|+c$

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