Mathematics

Write a value of 
$$\int { { e }^{ 2{ x }^{ 2 }+\ln { x }  } } dx$$


SOLUTION
$$I=\displaystyle\int{{e}^{2{x}^{2}+\ln{x}}dx}$$

$$=\displaystyle\int{{e}^{2{x}^{2}}{e}^{\ln{x}}dx}$$

$$=\displaystyle\int{{e}^{2{x}^{2}}x\,dx}$$

Let $$t={x}^{2}\Rightarrow\,dt=2x\,dx$$

$$=\displaystyle\int{{e}^{2t}dt}$$

$$=\dfrac{{e}^{2t}}{2}+c$$

$$=\dfrac{{e}^{2{x}^{2}}}{2}+c$$    .....where $$t={x}^{2}$$
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Subjective Medium Published on 17th 09, 2020
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