Mathematics

# Integrate the following function:$\displaystyle \int { \dfrac { { d }^{ 2 } }{ { dx }^{ 2 } } } \left( \sin ^{ -1 }{ x } \right) dx$

$(1-x^2)^{\frac{-1}{2}}+c$

##### SOLUTION
Consider, $\displaystyle I= \int { \cfrac { { d }^{ 2 } }{ d{ x }^{ 2 } } \left( \sin ^{ -1 }{ x } \right) dx }$

$\displaystyle I= \int { \cfrac { d }{ dx } \left( \cfrac { d }{ dx } \left( \sin ^{ -1 }{ x } \right) \right) dx }$

$\displaystyle I= \int { \cfrac { d }{ dx } \left( \cfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } \right) dx }$

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

$\displaystyle I=\int { \cfrac { x }{ { \left( 1-{ x }^{ 2 } \right) }^{ 3/2 } } dx }$

$\displaystyle \because 1-{ x }^{ 2 }=t;\quad -2xdx=dt$

$\displaystyle I=\int { \cfrac { -dt }{ 2{ t }^{ 3/2 } } } =\cfrac {- 1 }{ 2 } \int { { t }^{ -3/2 }dt }$

$\displaystyle I=\cfrac { -1 }{ 2 } \left( \cfrac { { t }^{ -1/2 } }{ -1/2 } \right) ={ \left( 1-{ x }^{ 2 } \right) }^{ -1/2 }+c$

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

#### Realted Questions

Q1 Subjective Medium
Evaluate the given integral.
$\int { { x }^{ 2 }\cos { x } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int_{0}^{\pi/4}{\dfrac{x.\sin x}{\cos^{3}x}dx}$ equals to :
• A. $\dfrac{\pi}{4}+\dfrac{1}{2}$
• B. $\dfrac{\pi}{4}$
• C. $\dfrac{\pi}{4}+1$
• D. $\dfrac{\pi}{4}-\dfrac{1}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate the integral
$\displaystyle \int_{1}^{e}\frac{(\ln x)^{3}}{x}dx$
• A. $e^{4}/4$
• B. $\displaystyle \frac{1}{4}({e}^{4}-1)$
• C. $e^{4}-1$
• D. $\dfrac{1}{4}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Let $[x]$ denote the largest integer not exceeding $x$ and $\left \{x\right \} = x - [x]$. Then
$\int_{0}^{2012} \dfrac {e^{\cos(\pi \left \{x\right \})}}{e^{\cos(\pi \left \{x\right \})} + e^{-\cos(\pi \left \{x\right \})}} dx$ is equal to.
• A. $0$
• B. $2012$
• C. $2012\pi$
• D. $1006$

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