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Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 44 Maths Textbook Solution.

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Answer: log x\left ( \frac{x^{2}}{2}+x \right )-\frac{x^{2}}{4}-x+c

Given:\int \left ( x+1 \right )log \: xdx

Hint: \int uvdx=u\int vdx-\int \left ( \frac{d}{dx}u\int vdx \right )dx


          \begin{aligned} &I=\int(x+1) \log x d x \\ &I=\left[\log x \int(x+1) d x-\int \frac{1}{x}\left(\frac{x^{2}}{2}+x\right) d x\right] \\ &I=\log x\left(\frac{x^{2}}{2}+x\right)-\int\left(\frac{x}{2}+1\right) d x \\ &=\log x\left(\frac{x^{2}}{2}+x\right)-\int \frac{x}{2} d x-\int d x \\ &=\log x\left(\frac{x^{2}}{2}+x\right)-\frac{x^{2}}{4}-x+c \end{aligned}

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