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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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