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

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

Solution:

\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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