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  • provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.9 question 6

provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise  18.9 question 6

Answers (1)

Answer: \frac{-1}{\left(1+e^{x}\right)}+c

Hint: Use substitution method to solve this integral.

Given: \int \frac{e^{x}}{\left(1+e^{x}\right)^{2}} d x

Solution:

        \begin{aligned} &\text { Let } I=\int \frac{e^{x}}{\left(1+e^{x}\right)^{2}} d x \\ &\text { Put } 1+e^{x}=t \Rightarrow e^{x} d x=d t \text { then } \end{aligned}

                  \begin{aligned} &I=\int \frac{1}{t^{2}} d t=\int t^{-2} d t \\ &=-\frac{t^{-2+1}}{-2+1}+c=\frac{t^{-1}}{-1}+\mathrm{c} \quad\left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}+c\right] \end{aligned}

                   \begin{aligned} &=-\frac{1}{t}+c \\ &=\frac{-1}{\left(1+e^{x}\right)}+c \quad\left[\because t=1+e^{x}\right] \end{aligned}

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