Given y= x^3/ (x^4 +10^5
We need to find the integral of y.
==> intg y = intg ( x^3/(x^4 +1)^5
Let us assume that u = x^4 + 1 ==> du = 4x^3 dx
Now we will substitute.
==> intg y = intg ( x^3 / u^5) * du/ 4x^3
We will reduce similar terms.
==> intg y = intg ( du/ 4u^5 )
= (1/4) intg u^-5 du
= (1/4) u^-4 / -4 + C
= (-1/16) u^-4 + C
= -1/16u^4 + C
Now we will substitute with u = x^4 +1
==> intg y = (-1/16)*(x^4+1)^4 + C
We'll solve the integral using substitution technique. We'll note x^4 + 1 = t.
We'll differentiate both sides:
4x^3dx = dt
x^3dx = dt/4
We'll write the integral in t:
Int dt/4t^5 = (1/4)Int t^-5dt
(1/4)Int t^-5dt = (1/4)*t^(-5+1)/(-5+1) +C
Int dt/4t^5 = -(1/16*t^4) + C
We'll substitute t = x^4 + 1
Int x^3*dx/(x^4+1)^5 = -1/16*(x^4 + 1)^4 + C