Mathematics

# $\displaystyle \int_{0}^{1}xe^{x}\ dx=$

$1$

##### SOLUTION
$\displaystyle\int_{0}^{1}{x{e}^{x}dx}$

Integrating by parts,

Let $u=x\Rightarrow\,du=dx$

$dv={e}^{x}dx\Rightarrow\,v={e}^{x}$

$\displaystyle\int_{0}^{1}{x{e}^{x}dx}=\left[x{e}^{x}\right]_{0}^{1}-\displaystyle\int_{0}^{1}{{e}^{x}dx}$

$=\left[{e}^{1}-0\right]_{0}^{1}-\left[{e}^{x}\right]_{0}^{1}$

$=e-\left[e-{e}^{0}\right]=e-e+1=1$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
Evaluate :
$\int { \log x } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Match the elements of List I with List II:
 List-I List-II A) $\displaystyle \int_{0}^{\frac{\pi}{2}}\log\sin{x}\>dx=$ 1) $4\pi$ B) $\displaystyle \int_{0}^{\frac{\pi}{2}}\log\tan x\>dx=$ 2) $-\dfrac{\pi}{2}\log_e2$ C) $\displaystyle\int_{0}^{\pi}x\log\sin x\>dx=$ 3) $-\dfrac{{\pi}^2}{2}\log_e2$ D) $\displaystyle\int_{-\pi}^{\pi}(x^{3}+x\cos x+\tan^{5}x+2)dx =$ 4) $0$
• A. $A-3, B-4, C-1, D-2$
• B. $A-1, B-4, C-2, D-3$
• C. $A-3, B-2, C-2, D-1$
• D. $A-2, B-4, C-3, D-1$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate:$\displaystyle \int \frac{(x^{2}+1)}{(x^{4}+x^{2}+1)}dx$
• A. $\tan^{-1}\dfrac{(x^{2}-1)}{\sqrt{3}x}+C$
• B. $\dfrac{1}{\sqrt{3}} \tan^{-1}\dfrac{(x^{2}-1)}{\sqrt{3}}+C$
• C. none of these
• D. $\dfrac{1}{\sqrt{3}} \tan^{-1}\dfrac{(x^{2}-1)}{\sqrt{3}x}+C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate:$\int _{ 1 }^{ 3 }{ \left( { x }^{ 2 }+3 \right)^{ 2 }dx }$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Hard
$\displaystyle I_1= \int_{0}^{\frac{\displaystyle \pi}{2}}ln(sin x) \:dx,I_2=\int_{-\frac{\displaystyle \pi}{4}}^{\frac{\displaystyle \pi}{4 }}\ln(\sin x+\cos x)\:dx$.Then
• A. $I_2=2I_1$
• B. $I_1=4I_2$
• C. $I_2=4I_1$
• D. $I_1=2I_2$