Mathematics

$\int \dfrac { e ^ { x } } { x } \left( x \cdot ( \log x ) ^ { 2 } + 2 \log x \right) d x$

SOLUTION
$\int { \cfrac { { e }^{ x } }{ x } \left( x{ \left( \log { x } \right) }^{ 2 }+2\log { x } \right) } dx\\ =\int { \cfrac { { e }^{ x } }{ x } \left( x{ \left( \log { x } \right) }^{ 2 }+\cfrac { 2\log { x } }{ x } \right) } dx\\ =\int { \cfrac { { e }^{ x } }{ x } \left( x{ \left( \log { x } \right) }^{ 2 } \right) } +\int { \cfrac { { e }^{ x } }{ x } } \cfrac { 2\log { x } }{ x } dx\\ ={ e }^{ x }{ \left( \log { x } \right) }^{ 2 }-\int { { e }^{ x } } \cfrac { 2\log { x } }{ x } dx+\int { { e }^{ x } } \cfrac { 2\log { x } }{ x } dx+C\\ ={ e }^{ x }{ \left( \log { x } \right) }^{ 2 }+C$

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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
The value of the integral $\int _ { 0 } ^ { \pi / 2 } \frac { 1 + 2 \cos x } { ( 2 + \cos x ) ^ { 2 } } d x$ is
• A. $\frac { 1 } { 4 }$
• B. $\frac { 1 } { 2 }$
• C. $\frac { -1 } { 4 }$
• D. $\frac { -1 } { 2 }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve $\int {\sqrt {\dfrac{{1 + x}}{{1 - x}}} }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium

Evaluate the following definite integral:

$\displaystyle\int_{0}^{\pi/4}\sin^{3}2t\cos 2t\ dt$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \int\frac{2x^{2}-5x+1}{x^{2}(x^{2}-1)}dx=$
• A. $\displaystyle \frac{1}{x}+\displaystyle log\left| \frac { x^{ 5 } }{ (x^{ 2 }-1)(x+1)^{ 2 } } \right| +$
• B. $\displaystyle \frac{1}{x}+\displaystyle \log\left| \frac { x^{ 5 } }{ (x^{ 2 }-1)(x+1)^{ 4 } } \right| +c$
• C. $\displaystyle \frac{1}{x}+\displaystyle \log\left| \frac { x^{ 5 } }{ (x^{ 2 }-1)(x+1) } \right| +c$
• D. $\displaystyle \frac{1}{x}+\displaystyle \log\left| \frac { x^{ 5 } }{ (x^{ 2 }-1)(x+1)^{ 3 } } \right| +c$

Evaluate $\displaystyle \int \frac{x^4 + 1}{x^6 + 1} dx$