Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • 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}

 

Posted by

infoexpert21

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Results 2024: Last 9 years’ CBSE Class 10, 12 result date and time; links to check score
anam-khan

CBSE Board exams twice a year from 2025, no plan for semesters: MoE
anam-khan

CBSE Class 12 board exams 2025 to have 50% competency-based questions; no change in 10th

Latest Question