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Need Solution for R.D. Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.3 Question 14 Maths Textbook Solution.

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Answer: \frac{e^{3 x}}{3}+e^{2 x}+e^{x}+c

Hint: \text { To solve this } \int x^{n} d x \text { formula }

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

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

\begin{aligned} &=\int\left(e^{2 x}+1+2 e^{x}\right) e^{x} d x \quad\left[(a+b)^{2}=a^{2}+b^{2}+2 a b\right] \\ &=\int\left(e^{3 x}+2 e^{2 x}+e^{x}\right) d x \end{aligned}

\begin{aligned} &=\int\left(e^{3 x}\right) d x+2 \int\left(e^{2 x}\right) d x+\int\left(e^{x}\right) d x \\ &=\frac{e^{3 x}}{3}+e^{2 x}+e^{x}+c \\ &=\frac{e^{3 x}+e^{2 x}+e^{x}}{3}+c \\ &=\frac{1}{3}\left(e^{x}+1\right)^{2}+c \end{aligned}

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