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Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 25 Maths Textbook Solution.

Answers (1)

Answer: \left(\frac{x}{\log 10}\right)[\log x-1]+c

Hint: Take

           \log x=\frac{\log x}{\log 10}

Given: \int \log _{10} x d x


\begin{aligned} &\int \log _{10} x d x=\int\left(\frac{\log x}{\log 10}\right) d x \\ &=\left(\frac{1}{\log 10}\right) \int \log x \cdot 1 d x \\ &=\left(\frac{1}{\log 10}\right)\left[(\log x) x-\int \frac{1}{x} x d x\right] \\ &=\left(\frac{1}{\log 10}\right)(x \log x-x)+c \\ &=x\left[\left(\frac{\log x}{\log 10}\right)-\left(\frac{1}{\log 10}\right)\right]+c \\ &=\left(\frac{x}{\log 10}\right)[\log x-1]+c \end{aligned}

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