Mathematics

# Integral of $\displaystyle f\left ( x \right )=\sqrt{1+x^{2}}$ with respect to $\displaystyle x^{2}$ is

$\displaystyle \frac{2}{3}\left ( 1+x^{2} \right )^{\tfrac 32}+k$

##### SOLUTION
Let $\displaystyle I=\int { f\left( x \right) d\left( { x }^{ 2 } \right) } =\int { 2x\sqrt { 1+{ x }^{ 2 } } dx }$

Substitute $t=\sqrt { 1+{ x }^{ 2 } } \Rightarrow dt=2xdx$

$\displaystyle \therefore I=\int { \sqrt { t } dt } =\dfrac { 2 }{ 3 } { t }^{ \dfrac { 3 }{ 2 } }+c$

$\displaystyle =\dfrac { 2 }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ \dfrac { 3 }{ 2 } }+k$

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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
Integrate: $\dfrac { 3 x ^ { 2 } } { x ^ { 6 } + 1 }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the definite integral   $\displaystyle \int_0^1\frac {dx}{\sqrt {1+x}-\sqrt x}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
By Simpson's rule, the value of $\displaystyle\int _{ -3 }^{ 3 }{ { x }^{ 4 }dy }$ by taking 6 sub-intervals, is
• A. $90$
• B. $80$
• C. $70$
• D. $98$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve  $\displaystyle\int {\dfrac{x}{{9 - 4{x^2}}}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Hard
Evaluate the given integral.
$\displaystyle \int { \cfrac { { e }^{ x }\left( 1+x \right) }{ \cos ^{ 2 }{ \left( x{ e }^{ x } \right) } } } dx$
• A. $2\log _{ e }{ \cos { \left( x{ e }^{ x } \right) } } +C$
• B. $\sec { \left( x{ e }^{ x } \right) +C }$
• C. $\tan { \left( x+{ e }^{ x } \right) } +C$
• D. $\tan { \left( x{ e }^{ x } \right) } +C$