Dgdrxrt

I have been told that "universal substitution always works", so I wanted to give it a try on this specific integral.

$int cos^4xdx$

For some reason it does not work. Please note that I solved this integral in a normal way, just don't understand why the universal substitution does not work.

$Bigg(t=tanx, cos^2x = frac11+t^2,dx=fracdt1+t^2 Bigg)$

$int cos^4xdx = int (cos^2x)^2dx = int Big(frac11+t^2 Big)^2 fracdt1+t^2 = int fracdt(1+t^2)(1+t^2)(1+t^2)$

$frac1(1+t^2)^3 = fracAt +B1+t^2 + fracCt+D(1+t^2)^2 + fracEt+F(1+t^2)^3$

$1 = (At+B)(t^4+2t^2+1)+(Ct+D)(1+t^2)+Et+F$

$1 = At^5 + 2At^3 + At + Bt^4 + 2Bt^2 + B + Ct^3 + Ct + Dt^2 + D + Et + F$

Now this boils down to six linear equations:

$0 = A$

$0 = B$

$0 = 2A + C$

$0 = 2B + D$

$0 = A + C + E$

$1 = B + D + F$

Which results in: $A = 0, B = 0, C = 0, D = 0, E = 0, F = 0$ making it unsolvable for me.

Any ideas what went wrong?

You made a mistake in solving your system of equations. The solution is
$$A = 0, B = 0, C = 0, D = 0, E = 0, F = 1$$

You could have seen this directly by observing that $frac1(1+t^2)^3$ is already in the desired form:
$$frac1(1+t^2)^3 = frac0t +01+t^2 + frac0t+0(1+t^2)^2 + frac0t+1(1+t^2)^3$$

You could have skipped this step and go directly to the "reduction formula"
$$intfracdt(t^2+1)^m=fract2(m-1)(t^2+1)^m-1+frac2m-32m-2intfracdt(t^2+1)^m-1$$
(or integration by parts if you are not familiar with this).

Your attempt at partial fraction decomposition is the problem; you can't remedy repeated factors like that. (Well, not usefully; see my comment.)

Instead you could cube $frac11+t^2=frac12ileft(frac1t-i-frac1t+iright)$ and simplify further. I don't recommend it, though. The simplest solution for the original problem is $$intcos^4 xdx=frac14int(1+cos 2x)^2dx=frac18int(3+4cos 2x+cos 4x)dx=frac3x+2sin 2x+frac14sin 4x8+C.$$

Another way using reduction formula

$$dfracd(cos^nxsin xdx=cos^n+1x-ncos^n-1x(cos^2x-1)=-(n-1)cos^n+1x+ncos^n-1x$$

Integrate both sides $$cos^nxsin x+K=-(n-1)I_n+1+nI_n-1$$

where $I_m=intcos^mx dx$ and $k$ is an arbitrary constant

Set $n=3,1$