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1 ∫ 0 E X 1 + E 2 X D X - Mathematics

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Question

\[\int\limits_0^1 \frac{e^x}{1 + e^{2x}} dx\]

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Solution

\[Let\ e^x = t . Then, e^x dx = dt\]
\[When\ x = 0, t = 1\ and\ x = 1, t = e\]
\[ \therefore I = \int_0^1 \frac{e^x}{1 + e^{2x}} d x\]
\[ \Rightarrow I = \int_1^e \frac{dt}{1 + t^2}\]
\[ \Rightarrow I = \left[ \tan^{- 1} x \right]_1^e \]
\[ \Rightarrow I = \tan^{- 1} e - \tan^{- 1} 1\]
\[ \Rightarrow I = \tan^{- 1} e - \frac{\pi}{4}\]

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Definite Integrals

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Chapter 20: Definite Integrals - Exercise 20.2 [Page 38]

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RD Sharma Mathematics [English] Class 12

Chapter 20 Definite Integrals
Exercise 20.2 | Q 6 | Page 38

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