Mathematics

# Evaluate : $\displaystyle\int \frac{\cos \sqrt{x}}{\sqrt{x}}dx.$

$\displaystyle 2\sin \sqrt{x}.$

##### SOLUTION
Let $\displaystyle I=\int \frac { \cos \sqrt { x } }{ \sqrt { x } } dx$

Put $\sqrt { x } =t\Rightarrow \dfrac { 1 }{ 2\sqrt { \left( x \right) } } dx=dt$

Therefore

$I=\displaystyle 2\int \cos tdt=2\sin t=2\sin \sqrt { x }$

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

#### Realted Questions

Q1 Single Correct Medium
$\int \dfrac{1-cos\, x}{cos\,x(1+cos\,x)}dx=$
• A. $log| sec\,x-tan\,x|-\,tan(\dfrac{x}{2})+c$
• B. $log| sec\,x+tan\,x|-2\,tan(\dfrac{x}{2})+c$
• C. $log| sec\,x-tan\,x|+\,tan(\dfrac{x}{2})+c$
• D. $log| sec\,x+tan\,x|+2\,tan(\dfrac{x}{2})+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int x^{2}(\log x)^{3}dx=$
• A. $\displaystyle x^{3}\left[ \frac { (\log x)^{ 3 } }{ 3 } +\frac { (\log x)^{ 2 } }{ 3 } +\frac { 2(\log x) }{ 9 } +\frac { 2 }{ 27 } \right] +c$
• B. $\displaystyle x^{3}\left[ \frac { (\log x)^{ 3 } }{ 3 } -\frac { (\log x)^{ 2 } }{ 27 } \right] +c$
• C. $\displaystyle x^{3}\left[ \frac { (\log x)^{ 3 } }{ 3 } +\frac { (\log x)^{ 2 } }{ 3 } +\frac { 2(\log x) }{ 9 } -\frac { 2 }{ 27 } \right] +c$
• D. $\displaystyle x^{3}\left[ \frac { (logx)^{ 3 } }{ 3 } -\frac { (logx)^{ 2 } }{ 3 } +\frac { 2(logx) }{ 9 } -\frac { 2 }{ 27 } \right] +c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
Which of the following definite integral(s) vanishes?
• A. $\displaystyle\int_0^{\displaystyle\frac{\pi}{2}}{\ln{(\cot{x})}dx}$
• B. $\displaystyle\int_0^{\displaystyle2\pi}{\sin^3{x}dx}$
• C. $\displaystyle \int _{ \displaystyle \frac { 1 }{ e } }^{ e }{\displaystyle \frac { dx }{ x{ \left( lnx \right) }^{\frac { 1 }{ 3 } } } }$
• D. $\displaystyle\int_0^{\displaystyle\pi}{\sqrt{\frac{1+\cos{2x}}{2}}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Prove that $\displaystyle \int _{-a}^{a} x^{3}\sqrt{a^{2}-x^{2}}dx=0$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
Evaluate: $\displaystyle \int { \dfrac { \cos { x } -\sin { x } }{ 1+\sin { 2x } } } dx$
• A. $\dfrac{1}{\sin x+\cos x}+C$
• B. $\dfrac{2}{\sin 2x+\cos x}+C$
• C. $-\dfrac{2}{\sin 2x+\cos x}+C$
• D. $-\dfrac{1}{\sin x+\cos x}+C$