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  • explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 4 maths

explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 4 maths

Answers (1)

Answer:\frac{2}{3}\left(1+e^{x}\right)^{\frac{3}{2}}+c

Hint: Use substitution method to solve this integral.

Given: \int \sqrt{1+e^{x}} \cdot e^{x} d x

Solution:

        Let I=\int \sqrt{1+e^{x}} \cdot e^{x} d x

        Put 1+e^{x}=t \Rightarrow e^{x} d x=d t \text { then }

        \begin{aligned} I &=\int \sqrt{t} d t=\int t^{\frac{1}{2}} d t \\ &=\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}+c=\frac{t^{\frac{3}{2}}}{\frac{3}{2}}+\mathrm{c} \quad\left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}+c\right] \\ &=\frac{2}{3}\left(1+e^{x}\right)^{\frac{3}{2}}+c \quad\left[\because t=1+e^{x}\right] \end{aligned}

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