I have 3 questions, and I cannot find method that actually solves them.

1) Integral [(4s+4)/([s^2+1]*([S-1]^3))]

2) Integral [ 2*sqrt[(1+cosx)/2]]

3) Integral [ 20*(sec(x))^4

Thanks in advance.

Explained here:

http://www.jiskha.com/display.cgi?id=1206288758

If you have trouble understanding this, then the best thing you can do is to replacethe problem by simpler problems and see if you can solve them.

E.g. try to integrate the function:

1/[(1+x)x^3]

Assuming that you are given these problems for homework, the purpose of solving the problems is not to get the right answer, it is for you to learn to solve integrals. So, it doesn't really matter what problems you solve to learn it. But trying to solve a problem for too long is an enormous waste of time.

I'm not quite sure about your explanation on Partial Fractions... And Liouville's Therom. We haven't done that in class... so...

Well, I think you just need to look at a simpler case. The algebra may sometimes obscue things. Also, you don't need to understand Lioville's theorem. Also, you don't need to stick to what you've been taught in class.

All you need to do is pick up a piece of paper and try to solve some problems. E.g. try to expand the following functions in partial fractions (using any method you like):

1/[x(1+x)]

1/[x^2 (1+x)]

1) A/x + B/(1+x)

2) A/x + B/x^2 + C/(1+x)

I think I'm right....

You now just need to solve for A, B, and C :)

That's my problem. When I tried solving for A, B, C, D, and E in that first problem, it didn't really work out right...

To solve the given integrals, we need to employ different methods for each question.

1) Integral [(4s+4)/([s^2+1]*([S-1]^3))]:
To find the integral of this function, we can use a partial fraction decomposition. First, decompose the denominator into its partial fractions: [s^2+1]*([S-1]^3) = (s^2 + 1) * [(s-1)^3]
So the decomposition becomes:
[4s+4]/([(s^2 + 1) * (s - 1)^3]) = A/(s + i) + B/(s - i) + C/(s - 1) + D/(s - 1)^2 + E/(s - 1)^3

Now, to find the values of A, B, C, D, and E, we need to solve for them. One way is to multiply both sides of the equation by the denominator [(s^2 + 1) * (s - 1)^3] and then substitute suitable values for s to eliminate terms and find the remaining unknowns.

Once we have found the values for A, B, C, D, and E, we can integrate each term separately. The integral of A/(s + i) can be found using the substitution method, whereas the integrals of B/(s - i), C/(s - 1), D/(s - 1)^2, and E/(s - 1)^3 can be found using the power rule.

2) Integral [2*sqrt[(1+cosx)/2]]:
To perform this integral, we use a trigonometric substitution. Let's substitute u = sin(x). Then, du = cos(x) dx. Using this substitution, the integral becomes:
Integral [2*sqrt[(1+u^2)/2]] du.

Now, we integrate [2*sqrt[(1+u^2)/2]] with respect to u. To do this, we can use a trigonometric identity to simplify the expression inside the square root. Remember that 1 - sin^2(x) = cos^2(x). So we can rewrite (1+u^2) as (1+sin^2(x)). This simplifies the integral to:
Integral [2*cos(x)] du.

Now, we can substitute back u = sin(x) and du = cos(x) dx to get the final result.

3) Integral [20*(sec(x))^4]:
To solve this integral, we can employ the power rule for integrals. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except for -1.

In this case, the integral of (sec(x))^4 is [(sec(x))^5]/5. However, we have a coefficient of 20 in front of our function. Therefore, we multiply our result by 20 to get the final answer.

I hope this explanation helps you to find the solutions to your integrals.