Mathematics

Evaluate: $$\displaystyle \overset{2}{\underset{0}{\int}} \dfrac{e^x }{e^{2x} + 1}dx$$.


SOLUTION
Now,
$$\displaystyle \overset{2}{\underset{0}{\int}} \dfrac{e^x }{e^{2x} + 1}dx$$
$$=\displaystyle \overset{2}{\underset{0}{\int}} \dfrac{e^x }{(e^{x})^2 + 1}dx$$
$$=\displaystyle \overset{2}{\underset{0}{\int}} \dfrac{d(e^x) }{(e^{x})^2 + 1}$$
$$=\left[\tan^{-1} e^x\right]_{x=0}^{x=2}$$
$$=\tan^{-1}e^2-\dfrac{\pi}{4}$$.
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Subjective Medium Published on 17th 09, 2020
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