Mathematics

# Write a value of $\displaystyle \int { \cfrac { 1 }{ 1+2{ e }^{ x } } } dx$

##### SOLUTION
$I=\displaystyle\int{\dfrac{1}{1+2{e}^{x}}dx}$

$=\displaystyle\int{\dfrac{{e}^{-x}}{{e}^{-x}\left(1+2{e}^{x}\right)}dx}$

$=\displaystyle\int{\dfrac{{e}^{-x}}{\left(2+{e}^{-x}\right)}dx}$

Let $t=2+{e}^{-x}\Rightarrow\,dt=-{e}^{-x}dx$

$=\displaystyle\int{\dfrac{-dt}{t}}$

$=-\log{\left(t\right)}+c$        .....where $c$ is the constant of integration.

$=-\log{\left(2+{e}^{-x}\right)}+c$     ...........where $t=2+{e}^{-x}$

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

#### Realted Questions

Q1 Single Correct Medium
Solve $\int { \left( { x }^{ 6 }+7{ x }^{ 5 }+6{ x }^{ 4 }+5{ x }^{ 3 }+4{ x }^{ 2 }+3x+1 \right) } { e }^{ x }dx=....+c$
• A. $\sum _{ i=0 }^{ 7 }{ { x }^{ i }{ e }^{ x } }$
• B. $\sum _{ i=0 }^{ 6 }{ i{ e }^{ x } }$
• C. $\sum _{ i=0 }^{ 6 }{ { (ex) }^{ i } }$
• D. $\sum _{ i=0 }^{ 6 }{ { x }^{ i }{ e }^{ x } }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int { \cfrac { dx }{ \sqrt { { x }^{ 2 }-6x+10 } } } =$?
• A. $\log { \left| x+\sqrt { { x }^{ 2 }-6x+10 } \right| } +C$
• B. $\log { \left| x-\sqrt { { x }^{ 2 }-6x+10 } \right| } +C$
• C. none of these
• D. $\log { \left| \left( x-3 \right) +\sqrt { { x }^{ 2 }-6x+10 } \right| } +C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Assertion & Reason Hard
##### ASSERTION

If $f,g$ and $h$ be continuous function on $[0,a]$ such that $f(x)=f(a-x),$ $g(x)=-g(a-x)$ and $3h(x)-4h(a-x)=5,$ then $\displaystyle \int _{ 0 }^{ a }{ f\left( x \right)g\left( x \right)h\left( x \right)dx } =0$

##### REASON

$\displaystyle \int _{ 0 }^{ a }{ f\left( x \right)g\left( x \right)dx } =0$

• A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\lim_\limits{n \to \infty}\left[ \dfrac{1}{n^2} \sec^2\dfrac{1}{n^2} +\dfrac{2}{n}\sec^2\dfrac{4}{n^2}............+\dfrac{1}{n} \sec^21 \right]$
• A. $\dfrac{1}{2} \sec 1$
• B. $\dfrac{1}{2} \text{cosec} 1$
• C. $\tan 1$
• D. $\dfrac{1}{2}\tan 1$

$\int \frac{2x^{2}}{3x^{4}2x} dx$