Question

Can someone break this down for me into steps? I have to evaluate the integral dx/ x(sqrt 5+4x)

Answer 1

I am thinking to rationalize the denominator first. Then decompose the fraction with a bystander sqrt(4+5x) (as in, 1/(x(4+5x)) * sqrt(4+5x) ). Distributing back in sqrt(4+5x) and integrating by term, one fraction can be taken down by some u-substitution and logarithm definition. The other seems to become u-substitution and power rule.

I can explain any part of this more clearly if you can't figure something out, and I'll see if there isn't an easier way to go about this that I am overlooking...

Answer 2

[-2tan^-1(sq rt(4x/5 +1)] / sq rt(5) + C ?

Answer 3

Did you use an identity for that one?

I'm looking at what wolfram had and both of your answers are super close except wolfram uses inverse hyperbolic tangent.

Answer 4

Hmm..I really don't know then. Not sure at all. Been trying to find something online similar. The example we had in class was slightly different and super easy. Was dx/ x (sqrt) 10x-1, which just uses the integral tables. But this has a different setup.

Answer 5

The minus sign may change it slightly
10x - 1 vs. 4 + 5x (5x + 4)
I was going with a way to do it without integral tables when I wrote my response, but I haven't done a lot with them recently. But I imagine if you used inverse tangent for the 10x -1 case, there is a counterpart inverse hyperbolic tangent for the sign difference.

Answer 6

Yeah trying to find a mirror problem in the text if possible

Answer 7

Can you solve this one? This is closest I found. dx/ (25-x^2) ^2

Answer 8

I found this general form:
http://www.sosmath.com/tables/integral/integ4/integ4.html
Integral #4 had the form you used listed as using a -b, but in this case b>0 so sqrt(-b) gives an imaginary number.
I think the alternate case can be turned into the hyperbolic tangent by taking the reciprocal and extracting the negative exponent

Answer 9

Roger

Answer 10

well i know that dx/ 25-x^2 is just sin^-1(x/5) + C but I'm having issues with that extra square

Answer 11

Nvm i got it

Answer 12

Sorry, a bit distracted. I was typing something and let it go to attend to something else.
Glad you were able to get it! And if you need another's help to look over this, feel free to bump the question! I haven't done so much with integral tables so someone else might be more knowledgeable. :)