Mathematics

# $\displaystyle \int_{0}^{\pi /2}\frac{\sin^{2}x}{\sin x+\cos x}dx=$

$\displaystyle \frac{1}{\sqrt{2}}l\mathrm{o}\mathrm{g}(\sqrt{2}+1)$

##### SOLUTION

$I=\int_{0}^{\dfrac{\pi}{2}}\dfrac{sin^2 x}{sin x + cos x}dx$
$I=\int_{0}^{\dfrac{\pi}{2}}\dfrac{cos^2}{sin x + cos x}dx$
$2I= \int_{0}^{\dfrac{\pi}{2}}\dfrac{1}{sin x + cos x}dx$
$2I=\int_{0}^{\dfrac{\pi}{2}}\dfrac{\left ( 1+ tan^2 \dfrac{x}{2}\right )}{-tan^2 \dfrac{x}{2}+ 2 tan \dfrac{x}{2}+1} dx$
$2I =\int_{0}^{\dfrac{\pi}{2}}\dfrac{2 d tan \dfrac{x}{2}}{(\sqrt{2})^2 \left ( tan \dfrac{x}{2} -1\right )^2}$
$=\dfrac{1}{2\sqrt{2}} log \left | \dfrac{\sqrt{2}+ tan^n (\dfrac{x}{2})-1}{\sqrt{2}(tan \dfrac{x}{2})+1} \right |$
$=\dfrac{1}{2\sqrt{2}} log \left | \dfrac{ tan^n (\dfrac{x}{2})+\sqrt{2}-1}{(\sqrt{2}+1)tan \dfrac{x}{2}} \right |\int_{0}^{\dfrac{\pi}{4}}$
$=\dfrac{1}{2}\sqrt{2} log ((\sqrt{2}+1)^2) = \dfrac{1}{\sqrt{2}} log (\sqrt{2}+1)$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Single Correct Hard
Evaluate: $\displaystyle\int \sqrt[3]{\frac{\sin ^{n}x}{\cos ^{n+6}x}}dx$
• A. $\displaystyle \frac{\tan ^{n/3}x}{n/3}$
• B. $\displaystyle \frac{\tan ^{n/3+1}x}{n/3}$
• C. $\displaystyle \frac{\cot ^{n/3+1}x}{n/3+1}$
• D. $\displaystyle \frac{\tan ^{n/3+1}x}{n/3+1}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $f(x)=\cfrac{x}{1+(\ln{x})(\ln{x})....\infty}\vee x\in [1,\infty )$ then $\int _{ 1 }^{ 2e }{ f\left( x \right) dx }$
• A. $\cfrac { { e }^{ 2 }+1 }{ 2 }$
• B. $\cfrac { { e }^{ 2 }-2e }{ 2 }$
• C. None of these
• D. $\cfrac { { e }^{ 2 }-1 }{ 2 }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int_{1}^{e}\log x \,dx =$________
• A. $e + 1$
• B. $e -1$
• C. $e + 2$
• D. $1$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int{\dfrac{x^{3}-1}{x^{3}+x}dx}$ equal to
• A. $x-\log x+\log(x^{2}+1)-\tan^{-1}x+c$
• B. $x+\log x+\dfrac{1}{2}\log(x^{2}+1)+\tan^{-1}x+c$
• C. $x+\log x-\dfrac{1}{2}\log(x^{2}+1)-\tan^{-1}x+c$
• D. $x-\log x+\dfrac{1}{2}\log(x^{2}+1)-\tan^{-1}x+c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\displaystyle \int {\frac{{\left( {{e^{2x}} - 1} \right)}}{{{e^{2x}} + 1}}} dx$
• A. $\log (e^x-e^{-x})+C$
• B. $\log (e^{2x}+e^{-2x})+C$
• C. $\log (e^{2x}-e^{-2x})+C$
• D. $\log (e^x+e^{-x})+C$