BASIC CALCULUS PLEASE HELP 10 POINTS TO BEST?

Question

here are a few problems i am stuck on in my homework. Answer as many as you would like, although i would prefer as much as possible.

integration sign (1+e^x)^2 divided by e^t multiplied by dt

integration sign (e^x+e^-x) divided by 2 with the entire numerator and denominator to the 3rd power(^3)

integration sign dx divided by x times square root(4x^2-1)

ANSWER ANY OR ALL HELP WILL BE APPRECIATED

Answer 1

let me start with the second one...

I think the easiest way to approach this is:

Int[((e^x+e^-x)/2)^3 dx]=

1/8 Int[e^3x +3 e^x +3 e^-x +e^-3x dx]

and this becomes:

1/8[1/3 e^3x +3 e^x -3 e^-x -1/3 e^-3x]

for the first one...I am presuming you mean for all terms to be of the same variable, i.e., all x or all t, I will use x

Int[(1+e^x)^2/e^x dx]=

Int[(1+2e^x+e^2x)/e^x dx]=

Int[e^-x +2 +e^x dx]=

-e^-x +2x +e^x