Mathematics

# Evaluate: $\displaystyle\int {{\dfrac{1} {{x^2} - x - 1}}} dx$

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

$=\displaystyle\int \dfrac{1}{x^2-2.x.\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{4}-1}dx$

$=\displaystyle\int \dfrac{1}{x^2-2.x.\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2-\dfrac{5}{4}}dx$

$=\displaystyle\int \dfrac{1}{\left(x-\dfrac{1}{2}\right)^2-\left(\dfrac{\sqrt{5}}{2}\right)^2}dx$

Let $x-\dfrac{1}{2}=u$

Here $a=\dfrac{\sqrt{5}}{2}$

$dx=du$

$=\displaystyle\int \dfrac{1}{u^2-a^2}du$ [by formula $\displaystyle\int \dfrac{dx}{x^2-a^2}=\dfrac{1}{2a}log\left|\dfrac{x-a}{x+a}\right|+c$]

$=\dfrac{1}{2a}log\left|\dfrac{u-a}{u+a}\right|+c$ (putting a and u)

$=\dfrac{1}{2\times \dfrac{\sqrt{5}}{2}}log\left|\dfrac{x-\dfrac{1}{2}-\dfrac{\sqrt{5}}{2}}{x-\dfrac{1}{2}+\dfrac{\sqrt{5}}{2}}\right|+c$

$=\dfrac{1}{\sqrt{5}} log\left|\dfrac{2x-1-\sqrt{5}}{2x-1+\sqrt{5}}\right|+c$.

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

#### Realted Questions

Q1 Single Correct Medium
Evaluate: $\displaystyle\int \frac { 1 } { ( x + 2 ) \sqrt { x + 1 } } d x$
• A. $\tan ^ { - 1 } ( \sqrt { x - 1 } ) + c$
• B. $2 \tanh ^ { - 1 } ( \sqrt { x + 1 } ) - c$
• C. $\tan ^ { - 1 } ( \sqrt { x + 1 } ) + c$
• D. $2 \tan ^ { - 1 } ( \sqrt { x + 1 } ) + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \dfrac {\sin x}{\sin 4x}dx$
• A. $\dfrac { 1 }{ 2\sqrt { 2 } } \log { \left| \dfrac { 1+\sqrt { 2\sin { x } } }{ 1-\sqrt { 2\sin { x } } } \right| } -\dfrac { 1 }{ 8 } \log { \left| \dfrac { 1+\sin { x } }{ 1-\sin { x } } \right| +C }$
• B. $\dfrac { 1 }{ 4\sqrt { 2 } } \log { \left| \dfrac { 1+\sqrt { 2\sin { x } } }{ 1-\sqrt { 2\sin { x } } } \right| } +\dfrac { 1 }{ 8 } \log { \left| \dfrac { 1+\sin { x } }{ 1-\sin { x } } \right| +C }$
• C. $\dfrac { 1 }{ 4\sqrt { 2 } } \log { \left| \dfrac { 1+\sqrt { 2\sin { x } } }{ 1-\sqrt { 2\sin { x } } } \right| } -\dfrac { 1 }{ 8 } \log { \left| \dfrac { 1+\sin { x } }{ 1-\sin { x } } \right| +C }$
• D. $\dfrac { 1 }{ 2\sqrt { 2 } } \log { \left| \dfrac { 1+\sqrt { 2\sin { x } } }{ 1-\sqrt { 2\sin { x } } } \right| } +\dfrac { 1 }{ 8 } \log { \left| \dfrac { 1+\sin { x } }{ 1-\sin { x } } \right| +C }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate $\displaystyle \int{\dfrac{dx}{{x}^{2}{\left(1+{x}^{4}\right)}^{\frac{3}{4}}}}$.

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Medium
$\displaystyle \int_{0}^{\infty }\left ( \cot ^{-1}x \right )^{2}dx= \frac{\pi}{k} \log 2$. Find the value of $k$.

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 $\displaystyle f(x)>0 \forall x \in R$