Mathematics

# $\int {\dfrac{{2x - 1}}{{2{x^2} + 2x + 1}}} dx =$

$\frac{1}{2}\ln \left| {2{x^2} + 2x + 1} \right| - 2{\tan ^{ - 1}}\left( {2x + 1} \right) + c$

##### SOLUTION
$\begin{array}{l} \frac { 1 }{ 2 } \left[ { \int { \frac { { 2x+1 } }{ { { x^{ 2 } }+x+\frac { 1 }{ 2 } } } dx-\int { \frac { 2 }{ { { x^{ 2 } }+x+\frac { 1 }{ 2 } } } } } dx } \right] \\ { x^{ 2 } }+x+\frac { 1 }{ 2 } =t,\, \, \, \, x+\frac { 1 }{ 2 } =u={ x^{ 2 } }+x+\frac { 1 }{ 4 } ={ u^{ 2 } } \\ \left( { 2x+1 } \right) dx=dt,\, \, \, \, x+\frac { 1 }{ 2 } =u \\ \frac { 1 }{ 2 } \left[ { \int { \frac { { dt } }{ t } -2\int { \frac { 1 }{ { { u^{ 2 } }+\frac { 1 }{ u } } } du } } } \right] \\ =\frac { 1 }{ 2 } \left[ { \ln { \left| t \right| } -2.\left( { \frac { 1 }{ { \frac { 1 }{ 2 } } } } \right) { { \tan }^{ -1 } }\left( { \frac { u }{ { \frac { 1 }{ 2 } } } } \right) } \right] +c \\ =\frac { 1 }{ 2 } \ln { \left| t \right| } -2{ \tan ^{ -1 } }2u+c \\ =\frac { 1 }{ 2 } \ln { \left| { { x^{ 2 } }+x+\frac { 1 }{ 2 } } \right| } -2{ \tan ^{ -1 } }\left( { 2x+1 } \right) +c \\ =\frac { 1 }{ 2 } \ln { \left| { 2{ x^{ 2 } }+2x+1 } \right| } -2{ \tan ^{ -1 } }\left( { 2x-1 } \right) +c \\ =\frac { 1 }{ 2 } \ln { \left| { 2{ x^{ 2 } }+2x+1 } \right| } -2{ \tan ^{ -1 } }\left( { 2x+1 } \right) +{ c_{ 1 } } \\ { c_{ 1 } }=c-\frac { 1 }{ 2 } \ln { 2 } \end{array}$

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

#### Realted Questions

Q1 Subjective Medium
Solve:
$\displaystyle \int e^{x}\left(\cfrac{1}{x}-\cfrac{1}{x^{2}}\right)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard

lf $f(x)$ is a polynomial satisfying$f(x)f(\frac{1}{x})=f(x) +f(\frac{1}{x})$, and$f(3)=82$, then $\displaystyle \int\frac{f(x)}{x^{2}+1}dx=$
• A. $x^{3}-x+2\tan^{-1}x+c$
• B. $\displaystyle \frac{1}{3}x^{3}-x+\tan^{-1}x+c$
• C. $\displaystyle \frac{1}{3}x^{3}+x+2\tan^{-1}x+c$
• D. $\displaystyle \frac{x^{3}}{3}-x+2\tan^{-1}x +c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate the following w.r.t.$\displaystyle \int x^2\left(1-\dfrac{2}{x}\right)^2dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle \int t^{3/2}+t^{1/2}dt$

Evaluate $\int { \dfrac { \tan ^{ 7 }{ \sqrt { x } } \sec ^{ 2 }{ \sqrt { x } } }{ \sqrt { x } } } dx$