Mathematics

Write a value of 
$$\int { { e }^{ x } } \sec{x} { \left( 1+\tan { x }  \right)  } dx$$


SOLUTION
$$I=\displaystyle\int{{e}^{x}\sec{x}\left(1+\tan{x}\right)dx}$$

$$\Rightarrow\,I=\displaystyle\int{{e}^{x}\sec{x}dx}+\displaystyle\int{{e}^{x}\sec{x}\tan{x}dx}$$

Consider $$\displaystyle\int{{e}^{x}\sec{x}\tan{x}dx}$$

Let $$u={e}^{x}\Rightarrow\,du={e}^{x}dx$$

$$dv=\sec{x}\tan{x}dx\Rightarrow\,v=\sec{x}$$

$$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ] $$......by parts formula.

$$\Rightarrow\,I=\displaystyle\int{{e}^{x}\sec{x}dx}+{e}^{x}\sec{x}-\displaystyle\int{{e}^{x}\sec{x}dx}+c$$

$$\therefore\,I={e}^{x}\sec{x}+c$$
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Subjective Medium Published on 17th 09, 2020
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