Question

What is integral of [x^2+1/{(x^2+4)^0.5}]?

Answer 1

`-ln|x+sqrt(x^2+4)|+x/2sqrt(x^2+4)+C`

Explanation:

I assume w.r.t. x...
`I=int(x^2+1)/sqrt(x^2+4)dx`

Substitute `x=2tanw`. Then `dx/(dw)=2sec^2w` and
`I=int(1+4tan^2w)/sqrt(4+4tan^2w)*2sec^2wdw`
`I=int(1+4tan^2w)/(2secw)*2sec^2wdw`
`I=int(1+4tan^2w)secwdw`
`I=intsecw+4secwtan^2wdw`
`I=intsecw+4secw(sec^2w-1)dw`
`I=int-3secw+4sec^3wdw`

The integral of `secxdx` is `ln|secx+tanx|`; see
https://socratic.org/questions/what-is-the-integral-of-sec-x

The integral of `sec^3xdx` is `1/2secxtanx+1/2ln|secx+tanx|`; see
https://socratic.org/questions/what-is-the-integral-of-sec-3-x

So
`I=-3ln|secw+tanw|+2secwtanw+2ln|secw+tanw|+C`
`I=-ln|secw+tanw|+2secwtanw+C`

Substitute back for our original variable `x`, using `tanw=x/2`
and `secw=sqrt(1+x^2/4)`:

`I=-ln|sqrt(1+x^2/4)+x/2|+xsqrt(1+x^2/4)+C`
`I=-ln(1/2|x+sqrt(x^2+4)|)+x/2sqrt(x^2+4)+C`
`I=-ln|x+sqrt(x^2+4)|+x/2sqrt(x^2+4)+C`
where we have absorbed the factor inside the log into the integration constant in order to make our expression similar to the question expression.