Recent content by scurty

You can use derivatives to show this (increasing/decreasing functions).

You completed the square wrong. You need to factor out a 9 before completing the square (x^2 coefficient has to be 1). Also, the fraction should have been 2/3, not 3/2.

Edit: Nevermind, I misread the problem. Some problems are tedious and you just have to work through them.

They are equivalent to each other. Plug their difference into WolframAlpha to see: ArcSinh[(2 + x)/Sqrt[17]]/2 - 1/2log(sqrt{(x+2)^2+17}/sqrt(17)+(x+2)/sqrt(17)) A good exercise for you would be to use the definition of arcsinh to write that form into your form. As a starting point...

That looks good to me. What did Wolfram give as an answer? Likely, you could manipulate that answer into the answer you gave.

I try to use u-substitution if I see a power that is one higher in the denominator than in the numerator. A more general approach would to be to look for derivatives in the numerator from what is found in the denominator. I always look for u-substitution first because they tend to be the easiest...

The step after you complete the square in the denominator. If you haven't learned hyberbolic trig functions, then use my suggestion. I have a feeling pasmith's substitution is a bit cleaner but there is nothing incorrect about using tangent.

The issue here is when you do the u substitution and plug in your values for u. You should have written ##\frac{1}{4}\displaystyle\int\frac{1}{\sqrt{u-17}\sqrt{u}}du## I wouldn't do u-substitution for this problem. Try doing trig substitution: ##x+2=\sqrt{17}\tan{\theta}##

"with respect to" It's just a shorthand notation that we use. Similar to WLOG (without loss of generality) and iff (if and only if).

Try writing a few terms out and notice what you can do with all the 3s that appear. Do you recall the Geometric Series?

I'm positive it is a typo, but there is a missing exponent in the second last step. I haven't worked with hyperbolic functions since college so I missed that neat shortcut. Nice work.

For respect to x: Perform a trig substitution (not theta, a different variable) by first completing the square under the square root. A nice simplification will occur. Then proceed as you normally would after a trig substitution to get the first answer provided in the image provided in the...

Since 5 and 3 are both prime, you only have 4 different possibilities of how the terms appear when you use the factor method. Perhaps the variable t is throwing you off? Start by letting t = 1 just so it's clearer to you and then go back and solve the equation with t. Also, you can only solve...

You made errors when using the chain rule (twice). I would go back and review that and revisit this problem after.

##e^0 = ...##