Some simple integral questions

[Emethyst]

Homework Statement

Evaluate the following integrals: a) (2+x)/(2sqrtx) on the interval [4,9] b) (e^-x + 1)/e^-x on the interval [0,ln2] c) 1/(sqrtx - sqrt(x-1)) on the interval [1,5]

Homework Equations

N/A

The Attempt at a Solution

My teacher has said that these questions should be like other basic antiderivative questions, but they seem to be stumping me, probably because of the fractions involved. In short I have no idea where to start with these questions, even though (supposedly) the answers are not that hard to find. If anyone can get me started on the right path for these questions here it would be greatly appreciated, thanks in advance.

 

[Cyosis]

For a) What is the derivative of \sqrt{x}. Do you see now what kind of substitution you can use?

b) Carry out the division. (a+b)c=a/c+b/c, it gives you a very simple integral.

c) Try multiplying by "1", with that I mean something of the form x/x, (x-1)/(x-1) etc. Do you see a good expression for "1"?

 

[Emethyst]

Thanks for the help cyosis, I was able to get b) and c). I'm still confused on a), however. I know that the derivative is 1/2x^-1/2, but I don't know how to bring the (2+x) into the anitderivative though, unless you mean substitution using another variable.

 

[Cyosis]

Make a substitution u=\sqrt{x}. You should recognize the derivative of this in the denominator of the integrand.