Mathematics

Evaluate the following integral
$$\int { \cfrac { { e }^{ x }+1 }{ { e }^{ x }+x }  } dx$$


SOLUTION
Let 
$$t={e}^{x}+x\Rightarrow\,dt=\left({e}^{x}+1\right)dx$$

$$\displaystyle\int{\dfrac{\left({e}^{x}+1\right)dx}{{e}^{x}+x}}$$

$$=\displaystyle\int{\dfrac{dt}{t}}$$

$$=\log{\left|t\right|}+c$$

$$=\log{\left|{e}^{x}+x\right|}+c$$ where $$t={e}^{x}+x$$
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Subjective Medium Published on 17th 09, 2020
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