Mathematics

# Solve $\int\limits_1^{-1} {\dfrac{d}{{dx}}ta{n^{ - 1}}\left( {\dfrac{1}{x}} \right)dx}$

##### SOLUTION

Consider the given integral.

$I=\int_{1}^{-1}{\dfrac{d}{dx}\left( {{\tan }^{-1}}\left( \dfrac{1}{x} \right) \right)}dx$

$I=\int_{1}^{-1}{\dfrac{1}{1+\dfrac{1}{{{x}^{2}}}}}dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ \dfrac{d}{dx}\left( {{\tan }^{-1}}x=\dfrac{1}{1+{{x}^{2}}} \right) \right]$

$I=\int_{1}^{-1}{\dfrac{{{x}^{2}}}{{{x}^{2}}+1}}dx$

$I=\int_{1}^{-1}{\dfrac{{{x}^{2}}+1-1}{{{x}^{2}}+1}}dx$

$I=\int_{1}^{-1}{1dx}-\int_{1}^{-1}{\dfrac{1}{{{x}^{2}}+1}}dx$

$I=\left[ x \right]_{1}^{-1}-\left[ {{\tan }^{-1}}\left( x \right) \right]_{1}^{-1}$

$I=\left[ -1-\left( 1 \right) \right]-\left[ {{\tan }^{-1}}\left( -1 \right)-{{\tan }^{-1}}\left( 1 \right) \right]$

$I=\left[ -2 \right]-\left[ {{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)-{{\tan }^{-1}}\left( \dfrac{\pi }{4} \right) \right]$

$I=\left[ -2 \right]-\left[ \dfrac{3\pi }{4}-\dfrac{\pi }{4} \right]$

$I=\left[ -2 \right]-\left[ \dfrac{\pi }{2} \right]$

$I=-2-\dfrac{\pi }{2}$

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

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int \cot x {dx}=$
• A. $\ln (\sin^2x) +C$
• B. $(\sin x) +C$
• C. None of these
• D. $\ln (\sin x) +C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle\int f(x)dx=g(x),$ then $\displaystyle\int f^{-1}(x)dx=$ _____________$+$c.
• A. $xf^{-1}(x)-g(f^{-1}(x))$
• B. $xf^{-1}(x)-g(f(x))$
• C. $x\cdot f^{-1}(x)$
• D. $x\cdot f(x)-g(f^{-1}(x))$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of $\displaystyle \int _{ -2 }^{ 2 }{ \left( x\cos { x } +\sin { x } +1 \right) }dx$ is
• A. $2$
• B. $0$
• C. $-2$
• D. $4$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle I= \int_{\pi /6}^{\pi /3}\frac{1}{1+\sqrt{\left ( \cot x \right )}}dx$
• A. $\displaystyle -\frac{\pi }{6}.$
• B. $\displaystyle \frac{\pi }{6}.$
• C. $\displaystyle -\frac{\pi }{12}.$
• D. $\displaystyle \frac{\pi }{12}.$

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.