Mathematics

# Solve :$\displaystyle \int \dfrac {dy}{e^{(x-1)y} -1 }$

##### SOLUTION
Consider, $I=\displaystyle\int { \dfrac { dy }{ { e }^{ \left( x-1 \right) y }-1 } }$

$\Rightarrow I= \displaystyle\int { \dfrac { dy }{ \dfrac { { e }^{ xy } }{ { e }^{ y } } -1 } }$

$=\displaystyle\int { \dfrac { { e }^{ y }dy }{ { e }^{ xy }-{ e }^{ y } } }$

${ e }^{ y }=t$ $\Rightarrow { e }^{ y }dy=dt$

$I= \displaystyle\int { \dfrac { dt }{ { e }^{ x }t-t } }$

$= \displaystyle\int { \dfrac { dt }{ t\left( { e }^{ x }-1 \right) } }$

$= \dfrac { 1 }{ \left( { e }^{ x }-1 \right) } \displaystyle\int { \dfrac { dt }{ t } }$

$= \dfrac { 1 }{ \left( { e }^{ x }-1 \right) } \left( n\left| t \right| +\left| n \right| c \right)$

$= c{ t }^{ \left( { e }^{ x }-1 \right) }$

$= c{ \left( { e }^{ y } \right) }^{ { e }^{ x }-1 }$

$\boxed{\Rightarrow c{ e }^{ y\left( { e }^{ x }-1 \right) }}$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Hard
Evaluate the following integral:
$\displaystyle \int { \cfrac { 1 }{ \sqrt { 5{ x }^{ 2 }-2x } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve
$\displaystyle \int \dfrac{dx}{(x^{1/2} + x^{1/3})}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Find $\int x^2 + a^2$ and evaluate $\int \dfrac{1}{3 + 2x + x^2}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate $\displaystyle \int \frac{{x{e^x}}}{{{{\left( {1 + x} \right)}^2}}} dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int \dfrac {dx}{(\sqrt {1 + x^{2}} - x)^{n}}(n\neq \pm 1) = \dfrac {1}{2} \left (\dfrac {z^{n + 1}}{n + 1} + \dfrac {z^{n - 1}}{n - 1}\right ) + O$
where
• A. $z = x - \sqrt {1 + x^{2}}$
• B. $z = \sqrt {1 + x^{2}} - x$
• C. $z = x - \sqrt {1 - x^{2}}$
• D. $z = x + \sqrt {1 + x^{2}}$