Question

Find the integral of √(x² +4) W.R.T x​

Answer 1

`(x)/(2) *\sqrt{x^(2) +4}` +
`(1)/(2)`*LN(|
`\frac{x+\sqrt{x^(2) +4} }{2}`|) +C

Explanation:

we will have to do a trig sub for this

use x=a*tanθ for sqrt(x^2 +a^2) where a=2

x=2tanθ, dx= 2 sec^2 (θ) dθ

this turns
`\int\limits {\sqrt{x^(2)+4 } } \, dx` into integral(sqrt( [2tanθ]^2 +4) * 2sec^2 (θ) )dθ

the sqrt( [2tanθ]^2 +4) will condense into 2sec^2 (θ) after converting tan^2(θ) into sec^2(θ) -1

then it simplifies into integral(4*sec^3 (θ)) dθ

you will need to do integration by parts to work out the integral of sec^3(θ) but it will turn into (1/2)sec(θ)tan(θ) + (1/2) LN(|sec(θ)+tan(θ)|) +C

then you will need to rework your functions of θ back into functions of x

tanθ will resolve back into
`(x)/(2)` (see substitutions) while secθ will resolve into
`\frac{\sqrt{x^(2) +4} }{2}`

sec(θ)=
`\frac{\sqrt{x^(2) +4} }{2}` is from its ratio identity of hyp/adj where the hyp. is
`\sqrt{x^(2) +4}` and adj is 2 (see tan(θ) ratio)

after resolving back into functions of x, substitute ratios for trig functions:

=
`(x)/(2) *\sqrt{x^(2) +4}` +
`(1)/(2)`*LN(|
`\frac{x+\sqrt{x^(2) +4} }{2}`|) +C