Calculus  Integrals
posted by David
I have 3 questions, and I cannot find method that actually solves them.
1) Integral [(4s+4)/([s^2+1]*([S1]^3))]
2) Integral [ 2*sqrt[(1+cosx)/2]]
3) Integral [ 20*(sec(x))^4
Thanks in advance.

Count Iblis
1) expand in partial fractions.
2) Using cos(2x) = 2 cos^2(x) 1 derive a formula for cos(1/2 x) in terms of cos(x). Express the integrand in terms of cos(1/2 x)
3) [Notation: cos = c, sin = s]
1/c^4 = (s^2 + c^2)/c^4 =
1/c^2 + s^2/c^4
1/c^2 yields a tangent when integrated
To integrate s^2/c^4 do partial integration:
s^2/c^4 = s (s/c^4)
Integral of s/c^4 is 1/3 1/c^3
So, we need to integrate
1/c^3
times the derivative of of s, i.e. 1/c^2, but that is tan(x)!
Note that the way to solve such problems is not to systematically work things out in detail at first, because then you would take too long to see that a method doesn't work.
Instead, you should reason like I just did, i.e. forget the details, be very sloppy, just to see if things works out and you get an answer in principle, even though you need to fill in the details.
If you get better at this, you can do the selection of what method to use in your head, you'll see it in just a few seconds when looking at an integral. You can then start to work out the solution for real on paper. 
David
Could you expand on #1 a bit more? I tried Partial Fractions, but I couldn't get a definite answer...

Count Iblis
I'll show you how to do it without solving any equations.
The function is up to a factor 4:
f(s) = (s+1)/{(s^2+1)*[(s1)^3]}
The partial fractions are precisely the singular terms when expanding around the singularities. So, let's examine the singularity at s = 1:
Put s = 1 + u and expand in powers of u, keeping only the singular terms:
f(1+u) = u^(3) (2+u)/[(u+1)^2+1] =
u^(3) (2+u)/(2+2u+u^2) =
u^(3) (2+u)/2 1/(1+u+u^2/2) = (use
1/(1+x) = 1x+x^2x^3+...)
u^(3) (2+u)/2 [1uu^2/2 +u^2+...]
=[u^(3) + u^(2)/2][1u+u^2/2+...]
u^(3) 1/2 u^(2) + nonsingular terms
This means that in the neighborhood of s = 1 we have:
f(s) = 1/(s1)^3  1/2 1/(s1)^2 +.. nonsingular terms
f(s) also has singularities at s = i and s =i. Around s = i, we have:
f(s) = a/(si) + nonsingular terms
Multiply both sides by (si) and take the limit s > i to find a:
a = (i+1)/[2i(i1)^3] =
1/2 1/(i1)^2
Around s = i, we have:
f(s) = b/(s+i) + nonsingular terms
Multiply both sides by (s+i) and take the limit s > i to find b:
b = (i+1)/[2i(i1)^3] =
1/2 1/[(i+1)^2]
The sum of all the singular terms of the expansions around the singular points is:
1/(s1)^3  1/2 1/(s1)^2 +
1/2 1/(i1)^2 1/(si)
1/2 1/[(i+1)^2] 1/(s+i) =
1/(s1)^3  1/2 1/(s1)^2 +
1/2 1/(s^2+1)
This is the desired partial fraction decomposition. The proof of why this works is a consequence of Liouville's theorem
http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)#Proof
Just consider the difference of f(s) and the sum of all the singular terms. The resulting function doesn't have any singularites and is bounded. So, by Liouville's theorem it is a constant. We know that f(s) and all the singular terms tend to zero, so that constant must be zero.
Respond to this Question
Similar Questions

CALCULUS 2!!! PLEASE HELP!!
I'm having trouble with this question on arc length: y=lnx, (squareroot)3/3 greater than or equal to x less than or equal to 1 It sounds as if you want the length of the y = ln x curve from x = sqrt(3)/3 (0.57735..) to 1. The formula … 
calc
how do you start this problem: integral of xe^(2x) There are two ways: 1) Integration by parts. 2) Differentiation w.r.t. a suitably chosen parameter. Lets do 1) first. This is the "standard method", but it is often more tedious than … 
calc asap!
can you help me get started on this integral by parts? 
Calculus
Find the volume of the solid whose base is the region in the xyplane bounded by the given curves and whose crosssections perpendicular to the xaxis are (a) squares, (b) semicircles, and (c) equilateral triangles. for y=x^2, x=0, … 
Calculus  Integrals
I have 3 questions, and I cannot find method that actually solves them. 1) Integral [(4s+4)/([s^2+1]*([S1]^3))] 2) Integral [ 2*sqrt[(1+cosx)/2]] 3) Integral [ 20*(sec(x))^4 Thanks in advance. 
Calculus
I have two questions, because I'm preparing for a math test on monday. 1. Use the fundamental theorem of calculus to find the derivative: (d/dt) the integral over [0, cos t] of (3/5(u^2))du I have a feeling I will be able to find … 
Calculus
Use the symmetry of the graphs of the sine and cosine functions as an aid in evaluating each definite integral. (a) Integral of sinx*dx from pi/4 to pi/4 (b) Integral of cosx*dx from pi/4 to pi/4 (c) Integral of cosx*dx from pi/2 … 
Calc
Evaluate the integral using any method: (Integral)sec^3x/tanx dx I started it out and got secx(1tan^2x)/tanx. I know I just have to continue simplifying and finding the integral, but I'm stuck on the next couple of steps. Also, I have … 
Calc 2
a. Integral (x^2)/(sqrt(1+(x^2))) Would I separate these two into 2 separate integrals? 
Math
Let A denote the portion of the curve y = sqrt(x) that is between the lines x = 1 and x = 4. 1) Set up, don't evaluate, 2 integrals, one in the variable x and one in the variable y, for the length of A. My Work: for x: integral[4,1] …