Mathematics

# Evaluate the following integral:$\int { \cfrac { \cot { x } }{ \sqrt { \sin { x } } } } dx$

##### SOLUTION
$\displaystyle\int{\dfrac{\cot{x}}{\sqrt{\sin{x}}}dx}$

$=\displaystyle\int{\dfrac{\cos{x}}{\sin{x}\sqrt{\sin{x}}}dx}$

Let $t=\sin{x}\Rightarrow\,dt=\cos{x}dx$

$=\displaystyle\int{\dfrac{dt}{t\sqrt{t}}}$

$=\displaystyle\int{\dfrac{dt}{{t}^{1+\frac{1}{2}}}}$

$=\displaystyle\int{\dfrac{dt}{{t}^{\frac{3}{2}}}}$

$=\displaystyle\int{{t}^{\frac{-3}{2}}dt}$

$=\dfrac{{t}^{\frac{-3}{2}+1}}{\dfrac{-3}{2}+1}+c$

$=\dfrac{{t}^{\frac{-1}{2}}}{\dfrac{-1}{2}}+c$

$=-\dfrac{2}{\sqrt{t}}+c$

$=-\dfrac{2}{\sqrt{\sin{x}}}+c$ where $t=\sin{x}$

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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
$\displaystyle \int cosec^{n}x\cdot cotxdx,(n\neq 0)=$
• A. $\displaystyle \dfrac{cosec^{n}x}{n}+c$
• B. $\dfrac{cosec^{n+1}x}{n+1}+c$
• C. $-\displaystyle \dfrac{cosec^{n+1}x}{n+1}+c$
• D. $-\displaystyle \dfrac{cosec^{n}x}{n}+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate the following integral:
$\int { \cfrac { x+5 }{ 3{ x }^{ 2 }+13x-10 } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int _{ 0 }^{ { \pi }/{ 8 } }{ { \cos }^{ 3 } } 4\theta d\theta$ is equal to:
• A. $\displaystyle \frac { 5 }{ 3 }$
• B. $\displaystyle \frac { 5 }{ 4 }$
• C. $\displaystyle \frac { 1 }{ 3 }$
• D. $\displaystyle \frac { 1 }{ 6 }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The evaluation of $\displaystyle\int{\frac{px^{\displaystyle p+2q-1}-qx^{\displaystyle q-1}}{x^{\displaystyle 2p+2q}+2x^{\displaystyle p+q}+1}dx}$ is
• A. $\displaystyle-\frac{x^p}{x^{\displaystyle p+q}+1}+C$
• B. $\displaystyle\frac{x^q}{x^{\displaystyle p+q}+1}+C$
• C. $\displaystyle\frac{x^p}{x^{\displaystyle p+q}+1}+C$
• D. $\displaystyle-\frac{x^q}{x^{\displaystyle p+q}+1}+C$

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