Question
Integrate the function $\sqrt{x^{2}+4 x-5}$
✓
Answer
$I=\sqrt{x^{2}+4 x-5} d x$
= $\int \sqrt{\left(x^{2}+4 x+4\right)-9} d x$
= $\int \sqrt{(x+2)^{2}-(3)^{2}} d x$
We know that,
$\Rightarrow \int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log |x+\sqrt{x^{2}-a^{2}}|+C$
$\Rightarrow \mathrm{I}=\frac{(\mathrm{x}+2)}{2} \sqrt{\mathrm{x}^{2}+4 \mathrm{x}-5}-\frac{9}{2} \log |(\mathrm{x}+2)+\sqrt{\mathrm{x}^{2}+4 \mathrm{x}-5}|+\mathrm{C}$
= $\int \sqrt{\left(x^{2}+4 x+4\right)-9} d x$
= $\int \sqrt{(x+2)^{2}-(3)^{2}} d x$
We know that,
$\Rightarrow \int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log |x+\sqrt{x^{2}-a^{2}}|+C$
$\Rightarrow \mathrm{I}=\frac{(\mathrm{x}+2)}{2} \sqrt{\mathrm{x}^{2}+4 \mathrm{x}-5}-\frac{9}{2} \log |(\mathrm{x}+2)+\sqrt{\mathrm{x}^{2}+4 \mathrm{x}-5}|+\mathrm{C}$
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