Mathematics

# If the value of the integral $\int_{1}^{2}{e^{x^{2}}}dx$ is $\alpha$, then the value of $\int_{e}^{e^4}{\sqrt{\log{x}}}dx$ is

$e^4-e-\alpha$

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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 Hard
The value of $\displaystyle\int _{ 0 }^{ \tfrac { \pi }{ 2 } }{ \frac { \cos ^{ \frac { 5 }{ 3 } }{ x } }{ \cos ^{ \frac { 5 }{ 3 } }{ x } +\sin ^{ \frac { 5 }{ 3 } }{ x } } dx }$ is
• A. $\dfrac { \pi }{ 2 }$
• B. $0$
• C. $\pi$
• D. $\dfrac { \pi }{ 4 }$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Hard
$\displaystyle \int \frac{dx}{\cos ^{3}x\sqrt{\left (\sin 2x \right )}}= \frac{\sqrt{2}}{5}\left ( t^{2}+5 \right )\sqrt{t}$. where $t=\tan x$.
If this is true enter 1, else enter 0.

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int_{}^{} {\dfrac{{{{(x - {x^5})}^{\cfrac{1}{5}}}}}{{{x^6}}}} dx$ is equal to
• A. $\dfrac{5}{{24}}{\left( {\dfrac{1}{{{x^4}}} - 1} \right)^{\cfrac{6}{5}}} + C$
• B. $\dfrac{5}{{24}}{\left( {1 - \dfrac{1}{{{x^4}}}} \right)^{\cfrac{6}{5}}} + C$
• C. none of these
• D. $- \dfrac{5}{{24}}{\left( {\dfrac{1}{{{x^4}}} - 1} \right)^{\cfrac{6}{5}}} + C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\int \sqrt {1 + \sin x}dx =$
• A. $\dfrac {1}{2}\left (\sin \dfrac {x}{2} + \cos \dfrac {x}{2}\right ) + c$
• B. $\dfrac {1}{2}\left (\sin \dfrac {x}{2} - \cos \dfrac {x}{2}\right ) + c$
• C. $2\sqrt {1 + \sin x} + c$
• D. $-2\sqrt {1 - \sin x} + c$

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