Mathematics

$$\displaystyle\int x^{3}\left ( \log x \right )^{2}dx=\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{8}x^{4}\log x+\frac{1}{4k}x^{4}.$$ Find the value of $$k$$


ANSWER

8


SOLUTION
$$\displaystyle I=\int x^{3}\left ( \log x \right )^{2}dx$$

$$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\int2\left ( \log x \right ).\frac{1}{x}.\frac{x^{4}}{4}dx$$

$$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{2}\int x^{3}\log dx$$

$$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{2}\left [ \frac{x^{4}}{4}\log x-\int \frac{1}{x}\frac{x^{4}}{4}dx \right ]$$

$$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{8}x^{4}\log x+\frac{1}{8}\int x^{3}dx$$

$$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{8}x^{4}\log x+\frac{1}{32}x^{4}dx$$
View Full Answer

Its FREE, you're just one step away


One Word Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
$$\displaystyle \int_{0}^{\pi/2}{(\sin x-\cos x).\log(\sin x+\cos x)dx}$$=
  • A. $$\dfrac{\pi}{2}$$
  • B. $$0$$
  • C. $$2$$
  • D. $$\dfrac{\pi}{4}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
Evaluate: $$\displaystyle\int \sec ^{4}x. \mathrm{cosec} ^{2}xdx$$
  • A. $$\displaystyle \frac{1}{3}t^{3}+t.$$
  • B. $$\displaystyle \frac{1}{2}t^{3}+2t-\frac{1}{t}.$$
  • C. $$\displaystyle \frac{1}{3}t^{3}-t-\frac{1}{t}.$$
  • D. $$\displaystyle \frac{1}{3}t^{3}+2t-\frac{1}{t}.$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
By using the properties of definite integrals, evaluate the integral   $$\displaystyle \int_0^{2\pi}\cos^5 x dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Hard
$$\int\frac{tanx+tan^3x}{1+tanx}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
$$\underset {n\rightarrow \infty}{lim}\dfrac{1^2+2^2+3^2+.....+n^2}{n^3}=.................$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer