Question

# $$\displaystyle \int { \frac { \sqrt { x } }{ \sqrt { { x }^{ 3 }+4 } } dx }$$ is equal to

A
23ln(2x3x34)+c
B
23ln(2x3+x34)+c
C
23ln(2x3x34)+c
D
None of these

Solution

## The correct option is B $$\displaystyle \frac { 2 }{ 3 } \ln { \left( \frac { 2 }{ \sqrt { { x }^{ 3 } } -\sqrt { { x }^{ 3 }-4 } } \right) } +c$$Given: $$\displaystyle \int { \frac { \sqrt { x } }{ \sqrt { { x }^{ 3 }+4 } } dx }$$Substitute $$\displaystyle { x }^{\tfrac 32 }=2\tan { \theta } \Rightarrow \frac { 3 }{ 2 } { x }^{\tfrac 12 }dx=2\sec ^{ 2 }{ \theta } d\theta$$$$\displaystyle \therefore I=\int { \dfrac { \dfrac { 4 }{ 3 } \sec ^{ 2 }{ \theta } d\theta }{ \sqrt { 4\tan ^{ 2 }{ \theta } +4 } } } =\frac { 2 }{ 3 } \int { \sec { \theta } d\theta }$$$$\displaystyle =\frac { 2 }{ 3 } \ln { \left( \sec { \theta } +\tan { \theta } \right) } +c=\frac { 2 }{ 3 } \ln { \left( \sqrt { \frac { { x }^{ 3 }-4 }{ 4 } } +\frac { { x }^{ \frac{3}{2} } }{ 2 } \right) } +c$$$$\displaystyle =\frac { 2 }{ 3 } \ln { \left( \frac { \sqrt { { x }^{ 3 } } +\sqrt { { x }^{ 3 }-4 } }{ 2 } \right) } +c=\frac { 2 }{ 3 } \ln { \left( \frac { { x }^{ 3 }-\left( { x }^{ 3 }-4 \right) }{ 2\left( \sqrt { { x }^{ 3 } } +\sqrt { { x }^{ 3 }-4 } \right) } \right) } +c$$$$\displaystyle =\frac { 2 }{ 3 } \ln { \left( \frac { 2 }{ \sqrt { { x }^{ 3 } } -\sqrt { { x }^{ 3 }-4 } } \right) } +c$$Mathematics

Suggest Corrections

0

Similar questions
View More

People also searched for
View More