{x²+x+1} dx × {1}{{x²+x+1}} dx =

Correct Answer is : (2x+14x2+x+1+38sinh⁡−12x+13)sinh⁡−1(2x+13)+C≤ft(2x+1/4{x²+x+1}+3/8^{-1}{2x+1}{{3}})^{-1}≤ft({2x+1}{{3}})+C(42x+1x2+x+1+83sinh−132x+1)sinh−1(32x+1)+C

≤ft(2x+1/4{x²+x+1}+3/8^{-1}{2x+1}{{3}})^{-1}≤ft({2x+1}{{3}})+C

2x+1/2^{-1}({x²+x+1})+≤ft(3/8^{-1}{2x+1}{{3}})²+C

2x+1/2≤ft(^{-1}{2x+1}{{3}})²+C

{x²+x+1} dx × {1}{{x²+x+1}} dx =

Correct Answer is : (2x+14x2+x+1+38sinh⁡−12x+13)sinh⁡−1(2x+13)+C≤ft(2x+1/4{x²+x+1}+3/8^{-1}{2x+1}{{3}})^{-1}≤ft({2x+1}{{3}})+C(42x+1x2+x+1+83sinh−132x+1)sinh−1(32x+1)+C

Test Index

TS EAMCET 6-Aug-2021 Shift 2 Question Paper

Section: Mathematics

Question : 76 of 160

Marks: +1, -0

\int\sqrt{x^2+x+1}

dx \times \int \frac{1}{\sqrt{x^2+x+1}}

dx

$x+C$
$\left(\frac{2x+1}{4}\sqrt{x^2+x+1}+\frac{3}{8}\sinh^{-1}\frac{2x+1}{\sqrt{3}}\right)\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+C$
$\frac{2x+1}{2}\sinh^{-1}(\sqrt{x^2+x+1})+\left(\frac{3}{8}\sinh^{-1}\frac{2x+1}{\sqrt{3}}\right)^2+C$
$\frac{2x+1}{2}\left(\sinh^{-1}\frac{2x+1}{\sqrt{3}}\right)^2+C$

Validate

Solution:

\because x^{2}+x+1=\left(x+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}

\therefore \int\sqrt{x^{2}+x+1}\,dx=\int\sqrt{\left(x+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}\,dx

= \frac{x+\frac{1}{2}}{2}\sqrt{x^{2}+x+1}+\frac{\frac{3}{2}}{2}\ln\left|\left(x+\frac{1}{2}\right)+\sqrt{x^{2}+x+1}\right|

= \frac{2x+1}{4}\sqrt{x^{2}+x+1}+\frac{3}{8}\ln\left|\frac{2x+1}{2}+\sqrt{x^{2}+x+1}\right|

and

\int\frac{1}{\sqrt{x^{2}+x+1}}\,dx=\int\frac{1}{\sqrt{\left(x+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}}\,dx

= \log\left|\left(x+\frac{1}{2}\right)+\sqrt{x^{2}+x+1}\right|

= \log\left|\frac{2x+1}{2}+\sqrt{x^{2}+x+1}\right|

\text{ Now, }\int\sqrt{x^{2}+x+1}\,dx \times \int\frac{1}{\sqrt{x^{2}+x+1}}\,dx

= \left[\frac{2x+1}{4}\sqrt{x^{2}+x+1}+\frac{3}{8}\log\left|\frac{2x+1}{2}+\sqrt{x^{2}+x+1}\right|\right]

\times \left[\log\left(\frac{2x+1}{2}+\sqrt{x^{2}+x+1}\right)\right]

=\left(\frac{2x+1}{4}\sqrt{x^{2}+x+1}+\frac{3}{8}\sinh^{-1}\frac{2x+1}{\sqrt{3}}\right)

\left(\sinh^{-1}\frac{2x+1}{\sqrt{3}}\right)+C

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