Mathematics

# $\displaystyle \int_{1}^{2}\frac{\mathrm{d}\mathrm{x}}{\sqrt{1+\mathrm{x}^{2}}}=$

$\displaystyle \log_{\mathrm{e}}(\frac{2+\sqrt{5}}{\sqrt{2}+1})$

##### SOLUTION

$\int_{1}^{2}\dfrac{dx}{\sqrt{1+x^2}}=\log |x+\sqrt{x^2+a}|+c$
$\int_{1}^{2}\dfrac{dx}{\sqrt{1+x^2}}=\log |x+\sqrt{x^2+1}|+c$
$=\log |2+\sqrt{5}|-\log |1+\sqrt{2}|$
$=\log \left | \dfrac{2+\sqrt{5}}{1+\sqrt{2}} \right |$
$\int_{1}^{2}\dfrac{dx}{\sqrt{1+x^2}}=\log \left | \dfrac{2+\sqrt{5}}{1+\sqrt{2}} \right |$

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

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle\int { \dfrac { \text{cosec} { x } }{ \cos ^{ 2 }{ \left( 1+\log { \tan { \dfrac { x }{ 2 } } } \right) } } dx }$ is equal to
• A. $\sin ^{ 2 }{ \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$
• B. $\sec ^{ 2 }{ \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$
• C. $-\tan { \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$
• D. $\tan { \left[ 1+\log { \tan { \dfrac { x }{ 2 } } } \right] } +C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following integral:
$\displaystyle\int^{\pi/2}_0\dfrac{dx}{(1+\cos^2x)}$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate:
$\int{\left( 3x+2 \right)\sqrt{10-4x-3{{x}^{2}}}}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate:
$\displaystyle \int \sqrt{5 - 2x + x^2}dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$