Mathematics

# $\int {{e^{3{{\log }_e}x}}.{{\left( {{x^4} + 1} \right)}^{ - 1}}dx = \_\_\_\_\_\_\_\_\_ + C.}$

$\frac{1}{4}\log \left( {{x^4} + 1} \right)$

##### SOLUTION
$I=\int { { e^{ 3\log_ex } }.{ { \left( { { x^{ 4 } }+1 } \right) }^{ -1 } }dx }$
$I=\int { { e^{ \log_ex^3 } }.{ { \left( { { x^{ 4 } }+1 } \right) }^{ -1 } }dx }$
$I=\int \dfrac{ { x^3 }}{{ { \left( { { x^{ 4 } }+1 } \right) } }}dx$

Let $t=x^4+1$
$dt=4x^3dx$

Therefore,
$I=\dfrac{1}{4}\int \dfrac{1}{t}dt$
$I=\dfrac{1}{4}\log(t)+C$

Put the value of $t$, we get
$I=\dfrac{1}{4}\log(x^4+1)+C$

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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:
$\displaystyle\int\limits_{-1}^{1}\sin^{2015}x.\cos^{2014}x\ dx$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int { \cfrac { 2{ x }^{ 3 } }{ \left( 4+{ x }^{ 8 } \right) } } dx=$?
• A. $\cfrac { 1 }{ 2 } \tan ^{ -1 }{ \cfrac { { x }^{ 4 } }{ 2 } } +C$
• B. $\cfrac { 1 }{ 2 } \tan ^{ -1 }{ { x }^{ 4 } } +C\quad$
• C. none of these
• D. $\cfrac { 1 }{ 4 } \tan ^{ -1 }{ \cfrac { { x }^{ 4 } }{ 2 } } +C\quad$

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Hard
If $f\left(x\right)=\dfrac{{e}^{x}}{1+{e}^{-x}},\,\,\,{I}_{1}=\displaystyle\int_{f\left(-a\right)}^{f\left(a\right)}{xg\left(x\left(1-x\right)\right)dx},\,\,\,{I}_{2}=\displaystyle\int_{f\left(-a\right)}^{f\left(a\right)}{g\left(x\left(1-x\right)\right)dx}$ then $\dfrac{{I}_{2}}{{I}_{1}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate: $\displaystyle \int_{0}^{1}\frac{x^{3}}{1+x^{8}} dx$
• A. $\displaystyle \frac{\pi}{4}$
• B. $\displaystyle \frac{\pi}{8}$
• C. $\displaystyle \frac{\pi}{2}$
• D. $\displaystyle \frac{\pi}{16}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
If $\displaystyle\int { \frac { \cos { 4x } +1 }{ \cot { x } -\tan { x } } } dx=A\cos { 4x } +B$; where $A$ & $B$ are constants, then
• A. $A=-1/4$ & $B$ may have any value
• B. $A=-1/2$ & $B=-1/4$
• C. $A=B=1/2$
• D. $A=-1/8$ & $B$ may have any value