Mathematics

# Evaluate the given integral.$\int { { e }^{ x } } \left( \cot { x } -co\sec ^{ 2 }{ x } \right) dx$

##### SOLUTION
$I=\displaystyle\int{{e}^{x}\left(\cot{x}-{cosec}^{2}{x}\right)dx}$

$I=\displaystyle\int{{e}^{x}\cot{x}\,dx}-\displaystyle\int{{e}^{x}{cosec}^{2}{x}\,dx}$

$u={e}^{x}\Rightarrow\,du={e}^{x}\,dx$

$dv={cosec}^{2}{x}\,dx\Rightarrow\,v=-\cot{x}$

$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ]$......by parts formula.

$\Rightarrow\,I=\displaystyle\int{{e}^{x}\cot{x}\,dx}-\left[-{e}^{x}\cot{x}-\displaystyle\int{-{e}^{x}\cot{x}dx}\right]+c$

$\Rightarrow\,I=\displaystyle\int{{e}^{x}\cot{x}\,dx}+{e}^{x}\cot{x}-\displaystyle\int{{e}^{x}\cot{x}dx}+c$

$\therefore\,I={e}^{x}\cot{x}+c$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
$\int {{3^x}\cos5x\,dx = }$
• A. $\dfrac{{{3^x}}}{{{{\left( {\log 3} \right)}^2} + 25}}\left[ {\left( {\log 3} \right).\cos5x - 5\sin5x} \right] + c$
• B. $\dfrac{{{3^x}}}{{{{\left( {\log 3} \right)}^2} + 25}}\left[ {5\cos5x - \left( {\log 3} \right).\sin5x} \right] + c$
• C. $\dfrac{{{3^x}}}{{{{\left( {\log 3} \right)}^2} + 25}}\left[ {5\cos5x + \left( {\log 3} \right).\sin5x} \right] + c$
• D. $\dfrac{{{3^x}}}{{{{\left( {\log 3} \right)}^2} + 25}}\left[ {\left( {\log 3} \right).\cos5x + 5\sin5x} \right] + c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
$\int { { x }^{ n }\log { x } dx }$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve $\int\limits_1^{-1} {\dfrac{d}{{dx}}ta{n^{ - 1}}\left( {\dfrac{1}{x}} \right)dx}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int { \dfrac { \text{cosec} { x } }{ \cos ^{ 2 }{ \left( 1+\log { \tan { \dfrac { x }{ 2 } } } \right) } } dx }$ is equal to
• A. $\sin ^{ 2 }{ \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$
• B. $\sec ^{ 2 }{ \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$
• C. $-\tan { \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$
• D. $\tan { \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
Evaluate the following:
$\int \cfrac{e^{3x}}{e^{3x}+1}.dx$
• A. $\dfrac{1}{3}ln(e^{3x}-1)+C$
• B. $\dfrac{1}{3}\ln(e^{4x}+1)+C$
• C. None of these
• D. $\dfrac{1}{3}ln(e^{3x}+1)+C$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020