Mathematics

Write a value of 
$$\displaystyle \int { \cfrac { \log { { x }^{ n } }  }{ x }  } dx$$


SOLUTION
$$I=\displaystyle\int{\dfrac{\log{{x}^{n}}}{x}dx}$$

$$=\displaystyle\int{\dfrac{n\log{x}}{x}dx}$$

$$=n\displaystyle\int{\dfrac{\log{x}}{x}dx}$$

Let $$t=\log{x}\Rightarrow\,dt=\dfrac{dx}{x}$$

$$=n\displaystyle\int{t\,dt}$$

$$=n\dfrac{{t}^{2}}{2}+c$$     .....where $$c$$ is the constant of integration.

$$=\dfrac{n}{2}{\left(\log{x}\right)}^{2}+c$$  .....where $$t=\log{x}$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
Solve:-
$$\displaystyle\int {\dfrac{{{e^x}(1 + x)}}{{{{\cos }^2}(x{e^x})}}} dx$$
  • A. $$I=2\tan \left( x{{e}^{x}} \right)+C$$
  • B. $$I=\tan \left( {{e}^{x}} \right)+C$$
  • C. None of these
  • D. $$I=\tan \left( x{{e}^{x}} \right)+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
Value of  $$\int_{\pi}^{2 \pi} [2 \, sin \, x] dx $$where [ ] represents the greatest integer function is 
  • A. $$-\pi$$
  • B. $$\frac { \pi }{3}$$
  • C. $$- 2\pi$$
  • D. $$\frac {5 \pi }{3}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate the following definite integrals :
$$\displaystyle \int _{0}^{\pi /2} \cos^2 x\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Hard
The integral $$\displaystyle \int_2^4\frac {log x^2}{log x^2+log (36-12x+x^2)}dx$$ is equal to
  • A. $$3$$
  • B. $$2$$
  • C. $$0$$
  • D. $$1$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let g(x) =$$\displaystyle \int_{0}^{x}f\left ( t \right )dt,$$ where f is a function
whose graph is show adjacently.
On the basis of above information, answer te following questions.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer