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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 15

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 15

Answers (1)

Answer:-\log|1+e^{-x}|+c

Hint: You must know about the integration rule of exponential function.

Given= \int \frac{1}{1+e^{x}}dx

Solution:

\begin{aligned} &\int \frac{1}{1+e^{x}} d x \\ &=\int \frac{e^{-x}}{e^{-x}\left(1+e^{x}\right)} d x \quad\quad\quad\quad\quad\quad\quad\quad\left [ multiply\: and \: divide\: by \: e^{x} \right ]\\ &=\int \frac{e^{-x}}{1+e^{-x}} d x \end{aligned}

Let e^{-x}+1=t                                                                              \left [ \frac{d}{dx}e^{-x}= -e^{-x} \right ]

\begin{aligned} &-\mathrm{e}^{-x} \mathrm{~d} \mathrm{x}=\mathrm{dt} \\ &I=\int \frac{-d t}{t} \\ &=-\log |t|+c \\ &=-\log \left|e^{-x}+1\right|+c\quad\quad\quad\quad\quad\quad \quad\left[\int \frac{1}{x} d x=\log x+c\right] \end{aligned}

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