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Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 14 Maths Textbook Solution.

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Answer: \frac{x^{n+1}}{n+1}\left[\log x-\frac{1}{n+1}\right]+c

Hint: \int u v d x=u \int v d x-\int \frac{d}{d x} u \int v d x

        Whereu=\log x, d v=x^{n} d x

Given: Let I=\int x^{n} \log x d x


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


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