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  • Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 21 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 21 Maths Textbook Solution.

Answers (1)

Answer:

              =\frac{x^{2}}{2}(\log x)^{2}-\frac{x^{2}}{2} \log x+\frac{x^{2}}{4}+c

Hint: Taking (\log x)^{2}as first function and 1-> Second function and integration

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

Given: 

          \int x(\log x)^{2} d x

Solution:

            I=\int x(\log x)^{2} d x

             Taking (\log x)^{2}as first function and x  as Second function and integrate

           \begin{aligned} &I=(\log x)^{2} \int x d x-\int\left\{\left[\frac{d}{d x}(\log x)^{2} \int x d x\right]\right\} d x \\ &=\frac{x^{2}}{2}(\log x)^{2}-\left[\int 2 \log x \frac{1}{x} \frac{x^{2}}{2} d x\right] \Rightarrow \frac{x^{2}}{2}(\log x)^{2}-\int x \log x d x \end{aligned}

Again integrating by parts, we obtain

           \begin{aligned} &I=\frac{x^{2}}{2}(\log x)^{2}-\left[\log x \int x d x-\int\left[\frac{d}{d x} \log x\right] \int x d x\right] d x \\ &=\frac{x^{2}}{2}(\log x)^{2}-\left[\frac{x^{2}}{2} \log x+\int \frac{1}{x} \frac{x^{2}}{2} d x\right] \\ &=\frac{x^{2}}{2}(\log x)^{2}-\frac{x^{2}}{2} \log x+\frac{1}{2} \int x d x \\ &=\frac{x^{2}}{2}(\log x)^{2}-\frac{x^{2}}{2} \log x+\frac{x^{2}}{4}+c \end{aligned}

 

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