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Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 22 Maths Textbook Solution.

Answers (1)

Answer: 2 e^{\sqrt{x}}(\sqrt{x}-1)+c

Hint: Put e^{\sqrt{x}}=t

Given: \int e^{\sqrt{x}} d x



              \begin{aligned} &e^{\sqrt{x}}=t \Rightarrow x=(\ln t)^{2} \Rightarrow d x=2 \frac{\ln t}{t} d t \\ &\therefore \int e^{\sqrt{x}} d x \Rightarrow 2 \int t \frac{\ln t}{t} d t=2 \int \ln t d t \end{aligned}

                [Now integrate by parts]

                  \Rightarrow 2\left[t \ln t-\int t d(\ln t)\right] \Rightarrow 2\left[t-\int t \frac{d t}{t}\right]

                  \begin{aligned} &=2\left[t \ln t-\int d t\right]=2[t \ln t-t+c] \\ &=2[t(\ln t-1)+c] \\ &=2 e^{\sqrt{x}}(\sqrt{x}-1)+c \end{aligned}

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