Question

integrate (1/(x^2+8x +25 )^0.5)

Answer 1

This is a hard one
x^2+8x+25=(x+4)^2+9
so use sub u=x+4 du=dx
New integral:
1/(u^2+9)^0.5

Answer 2

hmmm ya i also reached at this point but what after this?

Answer 3

thinking of the next substitution, maybe some trig function

Answer 4

hmm yes i think that would be tan but how and why

Answer 5

I know! Recall that tan^2(x)+1=sec^2(x)

Answer 6

so we need to make 9tan^2(x) from u to get 9(tan^2(x)+1)

Answer 7

let u=3tan(z)
du=3sec^2(z) dz

Answer 8

So the integral will be:
3sec^2(z)/3sec(z)=sec(z)

Answer 9

What the integral of secz? log(tanz+secz) (just looked it up)

Answer 10

we have u=3tanz expressing z=tan^-1(u/3)

Answer 11

thanks man.. well pretty much hit the answer but can u explain this 1 step let u=3tan(z) du=3sec^2(z) dz.. why we suppose 3 tan(z) why not just tan

Answer 12

because if it is 3tan than (3tan)^2=9tan^2(x) so we can factor out 9 to get 9(tan^2(x)+1)

Answer 13

ohh so just that was the reason to put 3?

Answer 14

yes

Answer 15

Hope it helped, now you just need to plug back the z and u to get x

Answer 16

do you know about hyperbolic functions ?

Answer 17

sinhx ?

Answer 18

and this is a elementary problem no need to think hard on this one

Answer 19

e^x+e^-x/2
how would that help?

Answer 20

\[\frac{d}{dx}\operatorname{arsinh}(x) =\frac{1}{\sqrt{x^{2}+1}}\]