Mathematics

# Evaluate the given integral.$\displaystyle \int { \cfrac { { e }^{ x }\left( 1+x \right) }{ \cos ^{ 2 }{ \left( x{ e }^{ x } \right) } } } dx$

$\tan { \left( x{ e }^{ x } \right) } +C$

##### SOLUTION
$I=\displaystyle\int{\dfrac{{e}^{x}\left(1+x\right)}{{\cos}^{2}{\left(x{e}^{x}\right)}}dx}$

Let $t=x{e}^{x}\Rightarrow\,dt=x{e}^{x}+{e}^{x}\,dx=\left(x+1\right){e}^{x}dx={e}^{x}\left(1+x\right)dx$

$\Rightarrow\,I=\displaystyle\int{\dfrac{dt}{{\cos}^{2}{t}}}$

$\Rightarrow\,I=\displaystyle\int{{\sec}^{2}{t}\,dt}$

Let $u=\tan{t}\Rightarrow\,du={\sec}^{2}{t}\,dt$

$\Rightarrow\,I=\displaystyle\int{du}$

$\Rightarrow\,I=u+c$

$\Rightarrow\,I=\tan{t}+c$     ............where $u=\tan{t}$

$\Rightarrow\,I=\tan{\left(x{e}^{x}\right)}+c$   ...........where $t=x{e}^{x}$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Medium
$\int\limits_{ - \pi /2}^{\pi /2} {\sin \left| x \right|} \,dx =$
• A.
• B. 2
• C. 4
• D. 1

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int { \frac { dx }{ x{ \left( { x }^{ 2 }+1 \right) }^{ 2 } } } =$
• A. $\displaystyle \ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } -\frac { 3 }{ 2\left( { x }^{ 2 }+1 \right) } +K$
• B. $\displaystyle -\ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } +\frac { 3 }{ 2\left( { x }^{ 2 }+1 \right) } +K$
• C. $\displaystyle -\ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } +\frac { 3 }{ 2\left( { x }+1 \right) } +K$
• D. $\displaystyle \ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } +\frac { 1 }{ 2\left( { x }^{ 2 }+1 \right) } +K$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate : $\displaystyle\int \left(\dfrac{1}{\log x} - \dfrac{1}{(\log x)^2} \right) . dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate:- $\int \dfrac{e^{2x} - 1}{e^{2x} + 1}$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.