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  • Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 45.

Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 45.

Answers (1)

Answer  : \frac{\log2}{1+\log 2}

 Hint   : use indefinite formula and the limit to solve this integral

 Given : \int_{1}^{2}\frac{1}{x(1+\log x)^2}dx

 Solution  : \int_{1}^{2}\frac{1}{x(1+\log x)^2}dx

put 1+\log x =t \Rightarrow \frac{ 1}{x}dx=dt \Rightarrow dx=x \; dt
when x=1 then t=1 when x=2 then t=1+log2
 

\begin{aligned} &\int_{1}^{2} \frac{1}{x(1+\log x)^{2}} d x=\int_{1}^{1+\log 2} \frac{1}{x \cdot t^{2}} x d t \\ &=\int_{1}^{1+\log 2} \frac{1}{t^{2}} d t \\ &=\int_{1}^{1+\log 2} t^{-2} d t \end{aligned}

\begin{aligned} &=\left[\frac{t^{-2+1}}{-2+1}\right]_{1}^{1+\log 2} \\ &=\left[\frac{t^{-1}}{-1}\right]_{1}^{1+\log 2} \\ &=-\left[\frac{1}{t}\right]_{1}^{1+\log 2} \\ &=-\left[\frac{1}{1+\log 2}-\frac{1}{1}\right] \\ &=1-\frac{1}{1+\log 2} \\ &=\frac{1+\log 2-1}{1+\log 2} \\ &=\frac{\log 2}{1+\log 2} \end{aligned}

 

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