Mathematics

# Solve  $\int { \cfrac { 1 }{ 9{ x }^{ 2 }+6x+10 } dx }$

##### SOLUTION
$\int { \cfrac { 1 }{ 9{ x }^{ 2 }+6x+10 } dx } =\int { \cfrac { 1 }{ 9{ x }^{ 2 }+6x+1+9 } dx }$
$=\int { \cfrac { 1 }{ { (3x+1) }^{ 2 }+9 } dx }$
$=\cfrac { 1 }{ 9 } \int { \cfrac { 1 }{ { \left( x+\cfrac { 1 }{ 3 } \right) }^{ 2 }+1 } dx }$
$=\cfrac { 1 }{ 9 } \int { \cfrac { 1 }{ { 1+\left( x+\cfrac { 1 }{ 3 } \right) }^{ 2 } } d\left( x+\cfrac { 1 }{ 3 } \right) }$
$=\cfrac { 1 }{ 9 } \tan ^{ -1 }{ \left( x+\cfrac { 1 }{ 3 } \right) } +C$

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

#### Realted Questions

Q1 Subjective Hard
Integrate the function    $\cfrac {x\cos^{-1}x}{\sqrt {1-x^2}}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$f'(x) =f(x), f(0) = 1$, then $\displaystyle \int \frac{dx}{f(x) + f(-x)}$
• A. $log (e^{2x} + 1) + C$
• B. $log (e^{x} + e^{-x}) + C$
• C. None
• D. $tan^{-1} (e^{x}) + C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle f(x)=\lim_{n\rightarrow \infty }(2x+4x^{3}+......+2^{n}x^{2n-1})\left ( 0<x<\frac{1}{\sqrt{2}} \right )$, then the value of $\displaystyle\int f(x) dx$ is equal to
$\textbf{Note}$: $c$ is the constant of integration.
• A. $\displaystyle \log\left ( \frac{1}{\sqrt{1-x^{2}}} \right )+c$
• B. $\displaystyle \log\sqrt{1-2x^{2}+x} + c$
• C. None of these
• D. $\displaystyle \log\left ( \frac{1}{\sqrt{1-2x^{2}}} \right )+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\int { \cfrac { 2dx }{ { x }^{ 2 }-1 } }$ equals:
• A. $\cfrac { 1 }{ 2 } \log { \left( \cfrac { x-1 }{ x+1 } \right) } +C$
• B. $\log { \left( \cfrac { x+1 }{ x-1 } \right) } +C$
• C. $\log { \left( \cfrac { x-1 }{ x+1 } \right) } +C$
• D. $\cfrac { 1 }{ 2 } \log { \left( \cfrac { x+1 }{ x-1 } \right) } +C$

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