<s>[INST] Different ways to solve $\int_0^1 \sqrt{1+x^2} dx $ So I have the task to solve this integral in $4$ different ways, but have solved it only with substitution. ($x=tg(t)$, $dx=\sec^2(t)dt$) and so on. Any advice on the other $3$ ways ? Thank you :)
 [/INST] IBP? $u = \sqrt{1+x^2}$ and $\mathrm{d}v = 1$ so $v = x$. Hence $$I = \int_0^1 \sqrt{1+x^2} \, \mathrm{d}x = \bigg[x\sqrt{1+x^2}\bigg]_0^1 - \int_0^1 \frac{x^2}{\sqrt{1+x^2}} \, \mathrm{d}x$$
Then $\int_0^1 \frac{x^2}{\sqrt{1+x^2}} \, \mathrm{d}x = \int_0^1 \frac{1+x^2}{\sqrt{1+x^2}} \, \mathrm{d}x - \int_0^1 \frac{1}{\sqrt{1+x^2}} \, \mathrm{d}x$. So $$I = \sqrt{2} - I + \int_0^1 \frac{1}{\sqrt{1+x^2}} \, \mathrm{d}x$$ or equivalently $$I = \frac{\sqrt{2}}{2} + \frac{1}{2}\text{arsinh} \,1$$

Alternatively, if you accept other substitutions $\sqrt{x^2 + 1} + x = t$ should work; this is known as Euler's substitution. 
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