Mathematics

# Evaluate:$\int { \cfrac { \cos { 2x } +2\sin ^{ 2 }{ x } }{ \sin ^{ 2 }{ x } } } dx$

##### SOLUTION
Given $\displaystyle\int{\dfrac{\cos{2x}+2{\sin}^{2}{x}}{{\sin}^{2}{x}}dx}$

$=\displaystyle\int{\dfrac{2{\cos}^{2}{x}-1+2{\sin}^{2}{x}}{{\sin}^{2}{x}}dx}$    $[\because \cos2x=2\cos^2-1]$

$=\displaystyle\int{\dfrac{2\left({\sin}^{2}{x}+{\cos}^{2}{x}\right)-1}{{\sin}^{2}{x}}dx}$

$=\displaystyle\int{\dfrac{2-1}{{\sin}^{2}{x}}dx}$ since $\left({\sin}^{2}{x}+{\cos}^{2}{x}=1\right)$

$=\displaystyle\int{\dfrac{1}{{\sin}^{2}{x}}dx}$

$=\displaystyle\int{{\csc}^{2}{x}dx}$

$=-\cot{x}+c$

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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 Hard
$\displaystyle \int_{\frac{5}{2}}^{5}\frac{\sqrt{(25-x^2)^3}}{x^4}\:dx$ is equal to
• A. $\displaystyle \frac{\pi }{6}$
• B. $\displaystyle \frac{2\pi }{3}$
• C. $\displaystyle \frac{5\pi }{6}$
• D. $\displaystyle \frac{\pi }{3}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate : $\int \dfrac { 1 } { x ^ { 2 } + 8 x + 20 } d x$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int { \frac { 1 }{ 1-\cos ^{ 4 }{ x } } dx } =-\frac { 1 }{ 2\tan { x } } +\frac { k }{ \sqrt { 2 } } \tan ^{ -1 }{ \left( \frac { \tan { x } }{ \sqrt { 2 } } \right) } +C$, where $k=$
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• B. $-1$
• C. $1$
• D. $\displaystyle \frac { 1 }{ 2 }$

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Hard
Evaluate the value of : $\displaystyle \int_0^{\frac{\pi}{2}} \dfrac{1}{1 + \cos x}dx$

The average value of a function f(x) over the interval, [a,b] is the number $\displaystyle \mu =\frac{1}{b-a}\int_{a}^{b}f\left ( x \right )dx$
The square root $\displaystyle \left \{ \frac{1}{b-a}\int_{a}^{b}\left [ f\left ( x \right ) \right ]^{2}dx \right \}^{1/2}$ is called the root mean square of f on [a, b]. The average value of $\displaystyle \mu$ is attained id f is continuous on [a, b].