Mathematics

# If $I=\int e^x\frac{(1-x)^2}{(1+x^2)^2}dx$ , then I =

$e^x\frac{1}{(1+x^2)}+C$

##### SOLUTION

$I=\int e^x\frac{(1-x)^2}{(1+x^2)^2}dx$
$I=\int e^x\frac{(1+x^2-2x)}{(1+x^2)^2}dx$
$I=\int e^x\frac{1+x^2}{(1+x^2)^2}dx$ $-$ $\int e^x\frac{2x}{(1+x^2)^2}dx$
$I=\int e^x\frac{1}{(1+x^2)}dx$ $-$ $\int e^x\frac{2x}{(1+x^2)^2}dx$
By integration by parts : $\int [f(x) g(x)]dx=f(x)\cdot \int g(x)dx-\int[f'(x) \int g(x) dx]dx$
By integration by  parts of first integral
$I=$ $\frac { 1 }{ 1+{ x }^{ 2 } } \int { e } ^{ x }-\int { \{ (\frac { d(\frac { 1 }{ 1+{ x }^{ 2 } } ) }{ dx } } )\int { { e }^{ x } } dx\} dx$   $-$ $\int e^x\frac{2x}{(1+x^2)^2}dx$
$I= e^x\frac{1}{(1+x^2)}$ $+$ $\int e^x\frac{2x}{(1+x^2)^2}dx$ $-$ $\int e^x\frac{2x}{(1+x^2)^2}dx$
$I= e^x\frac{1}{(1+x^2)}$ $+$ $C$

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

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int\frac{3x-4}{\sqrt{2x^{2}+4x+5}}dx=$
• A. $\displaystyle \frac{3}{2}\displaystyle \sqrt{2x^{2}+4x+5}+\frac{7}{\sqrt{2}}\sin h^{-1}(\frac{\sqrt{2}(x+1)}{\sqrt{3}})-c$
• B. $\displaystyle \frac{2}{3}\displaystyle \sqrt{2x^{2}-4x-5}+\frac{7}{\sqrt{2}}\sin h^{-1}(\frac{\sqrt{2}(x+1)}{\sqrt{3}})+c$
• C. $\displaystyle \frac{3}{2}\displaystyle \sqrt{2x^{2}+4x+5}-\frac{7}{\sqrt{2}}\sin h^{-1}(\frac{\sqrt{2}(x-1)}{\sqrt{3}})-c$
• D. $\displaystyle \frac{3}{2}\displaystyle \sqrt{2x^{2}+4x+5}-\frac{7}{\sqrt{2}}\sin h^{-1}(\frac{\sqrt{2}(x+1)}{\sqrt{3}})+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the definite integral $\displaystyle \int_0^{\tfrac {\pi}{2}}\cos 2x dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The value of $\displaystyle\int _{ 0 }^{ \infty }{ \dfrac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right) } }$ is
• A. $\dfrac { \pi }{ 20 }$
• B. $\dfrac { \pi }{ 40 }$
• C. $\dfrac { \pi }{ 80 }$
• D. $\dfrac { \pi }{ 60 }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Find the derivative of $\dfrac{e^{x}}{\sin x}$.
• A. $e^x\text{cosec }x[\cot x+1]$
• B. $e^x\sec x[\cot x+1]$
• C. $e^x\sec x[\cot x-1]$
• D. $e^x\text{cosec }x[1-\cot x]$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$