Mathematics

# Integrate $\displaystyle \int {\sqrt {1 + {x^2}} } dx$ is equal to:

##### SOLUTION
$= \mathop \smallint \nolimits^ {\sec ^2}\left( u \right)\sqrt {{{\tan }^2}\left( u \right) + 1} {\mkern 1mu} {\rm{d}}u$
$= \mathop \smallint \nolimits^ {\sec ^3}\left( u \right){\mkern 1mu} {\rm{d}}u$

$=\dfrac { { \sec \left( u \right) \tan \left( u \right) } }{ 2 } +\dfrac { 1 }{ 2 } \int { sec\left( u \right) } { { d } }u$

$\int {\sec \left( u \right)du}$
$= \ln \left( {\tan \left( u \right) + \sec \left( u \right)} \right)$
$\dfrac{{\sec \left( u \right)\tan \left( u \right)}}{2} + \dfrac{1}{2}\int {\sec \left( u \right)du}$
$= \dfrac{{\ln \left( {\tan \left( u \right) + \sec \left( u \right)} \right)}}{2} + \dfrac{{\sec \left( u \right)\tan \left( u \right)}}{2}$
$\mathop \smallint \nolimits^ \sqrt {{x^2} + 1} {\mkern 1mu} {\rm{d}}x$

$= \dfrac{{\ln \left( {\left| {\sqrt {{x^2} + 1} + x} \right|} \right)}}{2} + \dfrac{{x\sqrt {{x^2} + 1} }}{2} + C$

$\dfrac{{ar\sinh \left( x \right)}}{2} + \dfrac{{x\sqrt {{x^2} + 1} }}{2} + C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Single Correct Medium

$\displaystyle \int_{0}^{\pi /2}\log(\sin 2x)dx=$
• A. $\displaystyle \frac{\pi}{2}$ log2
• B. $\displaystyle \frac{\pi}{3}$ log2
• C. $\pi$ log2
• D. $-\displaystyle \frac{\pi}{2}$ log2

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int _{ a }^{ b }{ \cfrac { \log { x } }{ x } } dx=.......\quad$ (where $a,b\in { R }^{ + }$)
• A. $\cfrac { 1 }{ 2 } \log { \left( ab \right) }$
• B. $\log { \left( \cfrac { b }{ a } \right) }$
• C. $2\log { \left( \cfrac { b }{ a } \right) }$
• D. $\cfrac { 1 }{ 2 } \log { \left( ab \right) } \log { \left( \cfrac { b }{ a } \right) }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Solve:
$\displaystyle \int_{0}^{\frac{2}{3}} \cfrac{d x}{4+9 x^{2}} \text { equals }$
• A. $\dfrac {\pi}{6}$
• B. $\dfrac {\pi}{12}$
• C. $\dfrac {\pi}{4}$
• D. $\dfrac {\pi}{24}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Calculate the following integrals.
$\displaystyle \int_{-\pi/2}^{\pi/2} \, | sin \, x | \, dx.$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.