Mathematics

Evaluate the following integral
$$\int { \cfrac { 10{ x }^{ 9 }+{ 10 }^{ x }\log _{ e }{ 10 }  }{ { 10 }^{ x }+{ x }^{ 10 } }  } dx\quad $$


SOLUTION
Let 
$$t={10}^{x}+{x}^{10}$$

$$\Rightarrow\,dt=\left({10}^{x}\log{10}+10{x}^{9}\right)dx$$

$$\displaystyle\int{\dfrac{10{x}^{9}+{10}^{x}\log{10}}{{10}^{x}+{x}^{10}}dx}$$

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

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

$$=\log{\left|{10}^{x}+{x}^{10}\right|}+c$$ where $$t={10}^{x}+{x}^{10}$$
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Subjective Medium Published on 17th 09, 2020
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