Mathematics

Evaluate: $$\displaystyle\int {\left( {1 - \dfrac{{\log \dfrac{1}{v}}}{{{v^2}}}} \right)} \,dv$$


SOLUTION

Consider given the given integration,

$$I=\int{1-\dfrac{\log \dfrac{1}{v}}{{{v}^{2}}}dv}$$

$$I=\int{1dv-\int{\dfrac{\log \dfrac{1}{v}}{{{v}^{2}}}dv}}=v-\int{\dfrac{\log \dfrac{1}{v}}{{{v}^{2}}}dv}$$        …..(1)

Let,

$$y=\int{\dfrac{\log \dfrac{1}{v}}{{{v}^{2}}}dv}$$

Put,

$$ t=\dfrac{1}{v} $$

$$ dt=\dfrac{-1}{{{v}^{2}}}dv $$

$$ dv=-{{v}^{2}}dt $$

$$ y=\int{\dfrac{-{{v}^{2}}\log t}{{{v}^{2}}}dv}=-\int{\left( \log t \right).1dt} $$

$$ =-\left[ \log t.t-\int{\dfrac{1}{t}.tdt} \right] $$

$$ =-t.\log t+t+C $$

$$ y=-\dfrac{1}{{{v}^{2}}}\log \dfrac{1}{{{v}^{2}}}+\dfrac{1}{{{v}^{2}}}+C $$

Put, the value of  in equation (1),we get

$$ I=v-\left( -\dfrac{1}{{{v}^{2}}}\log \dfrac{1}{{{v}^{2}}}+\dfrac{1}{{{v}^{2}}}+C \right) $$

$$ I=v+\dfrac{1}{{{v}^{2}}}\log \dfrac{1}{{{v}^{2}}}-\dfrac{1}{{{v}^{2}}}-C $$


Hence, this is the answer,

 

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 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
If $$ \displaystyle    {I}_{ {n}}=\int^{\pi/2 }_{\pi/4}( {T} {a} {n}\theta)^{- {n}}. {d}\theta $$ for $$( {n}>1)$$ 
then $$I_{n}+I_{n+2} = ?$$
  • A. $$\displaystyle \frac{-1}{\mathrm{n}+1}$$
  • B. $$\displaystyle \frac{1}{\mathrm{n}-1}$$
  • C. $$\displaystyle \frac{-1}{\mathrm{n}-1}$$
  • D. $$\displaystyle \frac{1}{\mathrm{n}+1}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate the following : $$\displaystyle\int \dfrac{1}{\sqrt{3x^{2}-8}}.dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Integrate:
$$\displaystyle \int (x^2+4x+3) dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Find the value of $$\displaystyle\int\limits_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}}|\sin x|\ dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Evaluate the following integral:
$$\displaystyle\int^{\pi/6}_0\dfrac{\cos x}{(3+4\sin x)}dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer