Mathematics

$$\displaystyle \int \sqrt{x^{2}+2x+5}\; dx$$ is equal to


ANSWER

$$\displaystyle =\frac{1}{2}\left ( x+1 \right )\sqrt{x^{2}+2x+5}+2\:log$$

$$\displaystyle \left | \left ( x+1 \right )+\sqrt{x^{2}+2x+5} \right |+C$$


SOLUTION
$$\displaystyle I=\int \sqrt{x^{2}+2x+5}dx$$
$$\displaystyle =\int \sqrt{\left ( x+1 \right )^{2}+4}dx$$
Put $$x+1=t$$
$$\Rightarrow dx=dt$$

$$\displaystyle I=\int \sqrt{t^2+2^2}$$
$$\displaystyle =\frac { 1 }{ 2 } t\sqrt { t^{ 2 } +2^{ 2 } } +\frac { 1 }{ 2 } .\left( 2 \right) ^{ 2 }log\left| t+\sqrt { t^{ 2 }+2^{ 2 } }  \right| +C$$

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

$$\displaystyle =\frac{1}{2}\left ( x+1 \right )\sqrt{x^{2}+2x+5}+2\:log \left | \left ( x+1 \right )+\sqrt{x^{2}+2x+5} \right |+C$$
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Single Correct Medium Published on 17th 09, 2020
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