Mathematics

# $\int { \dfrac { 1 }{ { \left( 3x+1 \right) }^{ 2 }+{ 3 }^{ 2 } } dx\quad }$$\dfrac { 1 }{ 3 } \times \dfrac { 1 }{ 3 } { tan }^{ -1 }\left( \dfrac { 3x+1 }{ 3 } \right) +c$

##### SOLUTION
$\displaystyle\int{\dfrac{dx}{{\left(3x+1\right)}^{2}+{3}^{2}}}$
Let $t=3x+1\Rightarrow\,dt=3\,dx$
$\Rightarrow\,dx=\dfrac{1}{3}dt$
$\Rightarrow\,\displaystyle\int{\dfrac{dx}{{\left(3x+1\right)}^{2}+{3}^{2}}}$
$=\dfrac{1}{3}\displaystyle\int{\dfrac{dt}{{t}^{2}+{3}^{2}}}$
$=\dfrac{1}{3}\dfrac{1}{3}{\tan}^{-1}{\dfrac{t}{3}}+c$ where $c$ is the constant of integration.
$=\dfrac{1}{9}{\tan}^{-1}{\left(\dfrac{3x+1}{3}\right)}+c$ where $t=3x+1$

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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
Find the integrals of the functions in Exercises 1 to 22
1. ${\sin ^3}\left( {2x + 1} \right)$
2. $\,{\sin ^3}x{\cos ^3}x$
3.$\frac{{\cos x - \sin x}}{{1 + \sin 2x}}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\int { \dfrac { dx }{ \sqrt { { x-x }^{ 2 } } } } ;\left( x>\dfrac { 1 }{ 2 } \right)$ is equal to
• A. $2{ sin }^{ -1 }\sqrt { X } +C$
• B. $2{ sin }^{ -1 }\left( 2x-1 \right) +C$
• C. ${ cos }^{ -1 }2\sqrt { x-{ x }^{ 2 } } +C$
• D. $C-2{ cos }^{ -1 }\left( 2x-1 \right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve:
$\displaystyle \int{(ax+b)^3}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\int \:x^{-\frac{2}{3}}\left(1+x^{\frac{1}{2}}\right)^{-\frac{5}{3}}dx$ is equal to
• A. $3\left(1+x^{-\frac{1}{2}}\right)^{-\frac{2}{3}}+c$
• B. $3\left(1+x^{\frac{1}{2}}\right)^{-\frac{2}{3}}+c$
• C. none of these
• D. $3\left(1+x^{-\frac{1}{2}}\right)^{-\frac{1}{3}}+c$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.